This page intentionally left blank FFIRS 12/15/2010 10:13:22 Page 1 Introduction to Real Analysis This page intentionally left blank FFIRS 12/15/2010 10:13:22 Page 3 INTRODUCTION TO REAL ANALYSIS Fourth Edition Robert G. Bartle Donald R. Sherbert University of Illinois, Urbana-Champaign John Wiley & Sons, Inc. FFIRS 12/15/2010 10:13:22 Page 4 VP & PUBLISHER PROJECT EDITOR MARKETING MANAGER MEDIA EDITOR PHOTO RESEARCHER PRODUCTION MANAGER ASSISTANT PRODUCTION EDITOR COVER DESIGNER Laurie Rosatone Shannon Corliss Jonathan Cottrell Melissa Edwards Sheena Goldstein Janis Soo Yee Lyn Song Seng Ping Ngieng This book was set in 10/12 Times Roman by Thomson Digital, and printed and bound by Hamilton Printing Company. The cover was printed by Hamilton Printing Company. Founded in 1807, John Wiley & Sons, Inc. has been a valued source of knowledge and understanding for more than 200 years, helping people around the world meet their needs and fulfill their aspirations. 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Upon completion of the review period, please return the evaluation copy to Wiley. Return instructions and a free of charge return shipping label are available at www.wiley.com/go/returnlabel. Outside of the United States, please contact your local representative. Library of Congress Cataloging-in-Publication Data Bartle, Robert Gardner, 1927Introduction to real analysis / Robert G. Bartle, Donald R. Sherbert. – 4th ed. p. cm. Includes index. ISBN 978-0-471-43331-6 (hardback) 1. Mathematical analysis. 2. Functions of real variables. I. Sherbert, Donald R., 1935- II. Title. QA300.B294 2011 515–dc22 2010045251 Printed in the United States of America 10 9 8 7 6 5 4 3 2 1 FDED01 12/08/2010 15:42:42 Page 5 A TRIBUTE This edition is dedicated to the memory of Robert G. Bartle, a wonderful friend and colleague of forty years. It has been an immense honor and pleasure to be Bob’s coauthor on the previous editions of this book. I greatly miss his knowledge, his insights, and especially his humor. November 20, 2010 Urbana, Illinois Donald R. Sherbert FDED02 12/08/2010 15:43:51 Page 6 To Jan, with thanks and love. FPREF 12/09/2010 14:45:10 Page 7 PREFACE The study of real analysis is indispensable for a prospective graduate student of pure or applied mathematics. It also has great value for any student who wishes to go beyond the routine manipulations of formulas because it develops the ability to think deductively, analyze mathematical situations and extend ideas to new contexts. Mathematics has become valuable in many areas, including economics and management science as well as the physical sciences, engineering, and computer science. This book was written to provide an accessible, reasonably paced treatment of the basic concepts and techniques of real analysis for students in these areas. While students will find this book challenging, experience has demonstrated that serious students are fully capable of mastering the material. The first three editions were very well received and this edition maintains the same spirit and user-friendly approach as earlier editions. Every section has been examined. Some sections have been revised, new examples and exercises have been added, and a new section on the Darboux approach to the integral has been added to Chapter 7. There is more material than can be covered in a semester and instructors will need to make selections and perhaps use certain topics as honors or extra credit projects. To provide some help for students in analyzing proofs of theorems, there is an appendix on ‘‘Logic and Proofs’’ that discusses topics such as implications, negations, contrapositives, and different types of proofs. However, it is a more useful experience to learn how to construct proofs by first watching and then doing than by reading about techniques of proof. Results and proofs are given at a medium level of generality. For instance, continuous functions on closed, bounded intervals are studied in detail, but the proofs can be readily adapted to a more general situation. This approach is used to advantage in Chapter 11 where topological concepts are discussed. There are a large number of examples to illustrate the concepts, and extensive lists of exercises to challenge students and to aid them in understanding the significance of the theorems. Chapter 1 has a brief summary of the notions and notations for sets and functions that will be used. A discussion of Mathematical Induction is given, since inductive proofs arise frequently. There is also a section on finite, countable and infinite sets. This chapter can used to provide some practice in proofs, or covered quickly, or used as background material and returning later as necessary. Chapter 2 presents the properties of the real number system. The first two sections deal with Algebraic and Order properties, and the crucial Completeness Property is given in Section 2.3 as the Supremum Property. Its ramifications are discussed throughout the remainder of the chapter. In Chapter 3, a thorough treatment of sequences is given, along with the associated limit concepts. The material is of the greatest importance. Students find it rather natural though it takes time for them to become accustomed to the use of epsilon. A brief introduction to Infinite Series is given in Section 3.7, with more advanced material presented in Chapter 9. vii FPREF 12/09/2010 14:45:10 viii Page 8 PREFACE Chapter 4 on limits of functions and Chapter 5 on continuous functions constitute the heart of the book. The discussion of limits and continuity relies heavily on the use of sequences, and the closely parallel approach of these chapters reinforces the understanding of these essential topics. The fundamental properties of continuous functions on intervals are discussed in Sections 5.3 and 5.4. The notion of a gauge is introduced in Section 5.5 and used to give alternate proofs of these theorems. Monotone functions are discussed in Section 5.6. The basic theory of the derivative is given in the first part of Chapter 6. This material is standard, except a result of Caratheodory is used to give simpler proofs of the Chain Rule and the Inversion Theorem. The remainder of the chapter consists of applications of the Mean Value Theorem and may be explored as time permits. In Chapter 7, the Riemann integral is defined in Section 7.1 as a limit of Riemann sums. This has the advantage that it is consistent with the students’ first exposure to the integral in calculus, and since it is not dependent on order properties, it permits immediate generalization to complex- and vector-values functions that students may encounter in later courses. It is also consistent with the generalized Riemann integral that is discussed in Chapter 10. Sections 7.2 and 7.3 develop properties of the integral and establish the Fundamental Theorem of Calculus. The new Section 7.4, added in response to requests from a number of instructors, develops the Darboux approach to the integral in terms of upper and lower integrals, and the connection between the two definitions of the integral is established. Section 7.5 gives a brief discussion of numerical methods of calculating the integral of continuous functions. Sequences of functions and uniform convergence are discussed in the first two sections of Chapter 8, and the basic transcendental functions are put on a firm foundation in Sections 8.3 and 8.4. Chapter 9 completes the discussion of infinite series that was begun in Section 3.7. Chapters 8 and 9 are intrinsically important, and they also show how the material in the earlier chapters can be applied. Chapter 10 is a presentation of the generalized Riemann integral (sometimes called the ‘‘Henstock-Kurzweil’’ or the ‘‘gauge’’ integral). It will be new to many readers and they will be amazed that such an apparently minor modification of the definition of the Riemann integral can lead to an integral that is more general than the Lebesgue integral. This relatively new approach to integration theory is both accessible and exciting to anyone who has studied the basic Riemann integral. Chapter 11 deals with topological concepts. Earlier theorems and proofs are extended to a more abstract setting. For example, the concept of compactness is given proper emphasis and metric spaces are introduced. This chapter will be useful to students continuing on to graduate courses in mathematics. There are lengthy lists of exercises, some easy and some challenging, and ‘‘hints’’ to many of them are provided to help students get started or to check their answers. More complete solutions of almost every exercise are given in a separate Instructor’s Manual, which is available to teachers upon request to the publisher. It is a satisfying experience to see how the mathematical maturity of the students increases as they gradually learn to work comfortably with concepts that initially seemed so mysterious. But there is no doubt that a lot of hard work is required on the part of both the students and the teachers. Brief biographical sketches of some famous mathematicians are included to enrich the historical perspective of the book. Thanks go to Dr. Patrick Muldowney for his photograph of Professors Henstock and Kurzweil, and to John Wiley & Sons for obtaining portraits of the other mathematicians. FPREF 12/09/2010 14:45:10 Page 9 PREFACE ix Many helpful comments have been received from colleagues who have taught from earlier editions of this book and their remarks and suggestions have been appreciated. I wish to thank them and express the hope that they find this new edition even more helpful than the earlier ones. November 20, 2010 Urbana, Illinois Donald R. Sherbert THE GREEK ALPHABET A B G D E Z H Q I K L M a b g d e z h u i k l m Alpha Beta Gamma Delta Epsilon Zeta Eta Theta Iota Kappa Lambda Mu N X O P P S T Y F X C V n j o p r s t y w x c v Nu Xi Omicron Pi Rho Sigma Tau Upsilon Phi Chi Psi Omega This page intentionally left blank FTOC 12/08/2010 15:45:54 Page 11 CONTENTS CHAPTER 1 PRELIMINARIES 1 1.1 1.2 1.3 Sets and Functions 1 Mathematical Induction 12 Finite and Infinite Sets 16 CHAPTER 2 THE REAL NUMBERS 23 2.1 2.2 2.3 2.4 2.5 The Algebraic and Order Properties of R 23 Absolute Value and the Real Line 32 The Completeness Property of R 36 Applications of the Supremum Property 40 Intervals 46 CHAPTER 3 SEQUENCES AND SERIES 54 3.1 3.2 3.3 3.4 3.5 3.6 3.7 Sequences and Their Limits 55 Limit Theorems 63 Monotone Sequences 70 Subsequences and the Bolzano-Weierstrass Theorem 78 The Cauchy Criterion 85 Properly Divergent Sequences 91 Introduction to Infinite Series 94 CHAPTER 4 LIMITS 102 4.1 4.2 4.3 Limits of Functions 103 Limit Theorems 111 Some Extensions of the Limit Concept 116 CHAPTER 5 CONTINUOUS FUNCTIONS 124 5.1 5.2 5.3 5.4 5.5 5.6 Continuous Functions 125 Combinations of Continuous Functions 130 Continuous Functions on Intervals 134 Uniform Continuity 141 Continuity and Gauges 149 Monotone and Inverse Functions 153 xi FTOC 12/08/2010 15:45:54 xii Page 12 CONTENTS CHAPTER 6 DIFFERENTIATION 161 6.1 6.2 6.3 6.4 The Derivative 162 The Mean Value Theorem 172 L’Hospital’s Rules 180 Taylor’s Theorem 188 CHAPTER 7 THE RIEMANN INTEGRAL 198 7.1 7.2 7.3 7.4 7.5 Riemann Integral 199 Riemann Integrable Functions 208 The Fundamental Theorem 216 The Darboux Integral 225 Approximate Integration 233 CHAPTER 8 SEQUENCES OF FUNCTIONS 241 8.1 8.2 8.3 8.4 Pointwise and Uniform Convergence 241 Interchange of Limits 247 The Exponential and Logarithmic Functions 253 The Trigonometric Functions 260 CHAPTER 9 INFINITE SERIES 267 9.1 9.2 9.3 9.4 Absolute Convergence 267 Tests for Absolute Convergence 270 Tests for Nonabsolute Convergence 277 Series of Functions 281 CHAPTER 10 THE GENERALIZED RIEMANN INTEGRAL 288 10.1 10.2 10.3 10.4 Definition and Main Properties 289 Improper and Lebesgue Integrals 302 Infinite Intervals 308 Convergence Theorems 315 CHAPTER 11 A GLIMPSE INTO TOPOLOGY 326 11.1 11.2 11.3 11.4 Open and Closed Sets in R 326 Compact Sets 333 Continuous Functions 337 Metric Spaces 341 APPENDIX A LOGIC AND PROOFS 348 APPENDIX B FINITE AND COUNTABLE SETS 357 FTOC 12/08/2010 15:45:54 Page 13 CONTENTS xiii APPENDIX C THE RIEMANN AND LEBESGUE CRITERIA 360 APPENDIX D APPROXIMATE INTEGRATION 364 APPENDIX E TWO EXAMPLES 367 REFERENCES 370 PHOTO CREDITS 371 HINTS FOR SELECTED EXERCISES 372 INDEX 395 This page intentionally left blank C01 12/08/2010 12:3:14 Page 1 CHAPTER 1 PRELIMINARIES In this initial chapter we will present the background needed for the study of real analysis. Section 1.1 consists of a brief survey of set operations and functions, two vital tools for all of mathematics. In it we establish the notation and state the basic definitions and properties that will be used throughout the book. We will regard the word ‘‘set’’ as synonymous with the words ‘‘class,’’ ‘‘collection,’’ and ‘‘family,’’ and we will not define these terms or give a list of axioms for set theory. This approach, often referred to as ‘‘naive’’ set theory, is quite adequate for working with sets in the context of real analysis. Section 1.2 is concerned with a special method of proof called Mathematical Induction. It is related to the fundamental properties of the natural number system and, though it is restricted to proving particular types of statements, it is important and used frequently. An informal discussion of the different types of proofs that are used in mathematics, such as contrapositives and proofs by contradiction, can be found in Appendix A. In Section 1.3 we apply some of the tools presented in the first two sections of this chapter to a discussion of what it means for a set to be finite or infinite. Careful definitions are given and some basic consequences of these definitions are derived. The important result that the set of rational numbers is countably infinite is established. In addition to introducing basic concepts and establishing terminology and notation, this chapter also provides the reader with some initial experience in working with precise definitions and writing proofs. The careful study of real analysis unavoidably entails the reading and writing of proofs, and like any skill, it is necessary to practice. This chapter is a starting point. Section 1.1 Sets and Functions To the reader: In this section we give a brief review of the terminology and notation that will be used in this text. We suggest that you look through it quickly and come back later when you need to recall the meaning of a term or a symbol. If an element x is in a set A, we write x2A and say that x is a member of A, or that x belongs to A. If x is not in A, we write x2 = A: If every element of a set A also belongs to a set B, we say that A is a subset of B and write AB or B A: 1 C01 12/08/2010 12:3:15 2 Page 2 CHAPTER 1 PRELIMINARIES We say that a set A is a proper subset of a set B if A B, but there is at least one element of B that is not in A. In this case we sometimes write A B: 1.1.1 Definition Two sets A and B are said to be equal, and we write A ¼ B, if they contain the same elements. Thus, to prove that the sets A and B are equal, we must show that AB and B A: A set is normally defined by either listing its elements explicitly, or by specifying a property that determines the elements of the set. If P denotes a property that is meaningful and unambiguous for elements of a set S, then we write fx 2 S : PðxÞg for the set of all elements x in S for which the property P is true. If the set S is understood from the context, then it is often omitted in this notation. Several special sets are used throughout this book, and they are denoted by standard symbols. (We will use the symbol :¼ to mean that the symbol on the left is being defined by the symbol on the right.) The The The The set set set set of of of of natural numbers N :¼ f1; 2; 3; . . .g, integers Z :¼ f0; 1; 1; 2; 2; . . .g, rational numbers Q :¼ fm=n : m; n 2 Z and n 6¼ 0g, real numbers R. The set R of real numbers is of fundamental importance and will be discussed at length in Chapter 2. 1.1.2 Examples (a) The set x 2 N : x2 3x þ 2 ¼ 0 consists of those natural numbers satisfying the stated equation. Since the only solutions of this quadratic equation are x ¼ 1 and x ¼ 2, we can denote this set more simply by {1, 2}. (b) A natural number n is even if it has the form n ¼ 2k for some k 2 N. The set of even natural numbers can be written f2k : k 2 N g; which is less cumbersome than fn 2 N : n ¼ 2k; k 2 N g. Similarly, the set of odd natural numbers can be written f2k 1 : k 2 N g: & Set Operations We now define the methods of obtaining new sets from given ones. Note that these set operations are based on the meaning of the words ‘‘or,’’ ‘‘and,’’ and ‘‘not.’’ For the union, it is important to be aware of the fact that the word ‘‘or’’ is used in the inclusive sense, allowing the possibility that x may belong to both sets. In legal terminology, this inclusive sense is sometimes indicated by ‘‘and/or.’’ C01 12/08/2010 12:3:15 Page 3 1.1 SETS AND FUNCTIONS 3 1.1.3 Definition (a) The union of sets A and B is the set A [ B :¼ fx : x 2 A or x 2 Bg: (b) The intersection of the sets A and B is the set A \ B :¼ fx : x 2 A and x 2 Bg: (c) The complement of B relative to A is the set AnB :¼ fx : x 2 A and x 2 = Bg: Figure 1.1.1 (a) A [ B (b) A \ B (c) AnB The set that has no elements is called the empty set and is denoted by the symbol ;. Two sets A and B are said to be disjoint if they have no elements in common; this can be expressed by writing A \ B ¼ ;. To illustrate the method of proving set equalities, we will next establish one of the De Morgan laws for three sets. The proof of the other one is left as an exercise. 1.1.4 Theorem If A, B, C are sets, then (a) AnðB [ C Þ ¼ ðAnBÞ \ ðAnC Þ, (b) AnðB \ C Þ ¼ ðAnBÞ [ ðAnC Þ. Proof. To prove (a), we will show that every element in AnðB [ C Þ is contained in both (AnB) and (AnC), and conversely. If x is in AnðB [ CÞ, then x is in A, but x is not in B [ C. Hence x is in A, but x is neither in B nor in C. Therefore, x is in A but not B, and x is in A but not C. Thus, x 2 AnB and x 2 AnC, which shows that x 2 ðAnBÞ \ ðAnC Þ. Conversely, if x 2 ðAnBÞ \ ðAnCÞ, then x 2 ðAnBÞ and x 2 ðAnCÞ. Hence x 2 A and both x 2 = B and x 2 = C. Therefore, x 2 A and x 2 = ðB [ CÞ, so that x 2 AnðB [ C Þ. Since the sets ðAnBÞ \ ðAnC Þ and AnðB [ C Þ contain the same elements, they are equal by Definition 1.1.1. Q.E.D. There are times when it is desirable to form unions and intersections of more than two sets. For a finite collection of sets {A1, A2, . . . , An}, their union is the set A consisting of all elements that belong to at least one of the sets Ak, and their intersection consists of all elements that belong to all of the sets Ak. This is extended to an infinite collection of sets {A1, A2, . . . , An, . . . } as follows. Their union is the set of elements that belong to at least one of the sets An. In this case we write 1 [ An :¼ fx : x 2 An for some n 2 N g: n¼1 C01 12/08/2010 12:3:16 4 Page 4 CHAPTER 1 PRELIMINARIES Similarly, their intersection is the set of elements that belong to all of these sets An. In this case we write 1 \ An :¼ fx : x 2 An for all n 2 N g: n¼1 Functions In order to discuss functions, we define the Cartesian product of two sets. 1.1.5 Definition If A and B are nonempty sets, then the Cartesian product A B of A and B is the set of all ordered pairs (a, b) with a 2 A and b 2 B. That is, A B :¼ fða; bÞ : a 2 A; b 2 Bg: Thus if A ¼ {1, 2, 3} and B ¼ {1, 5}, then the set A B is the set whose elements are the ordered pairs ð1; 1Þ; ð1; 5Þ; ð2; 1Þ; ð2; 5Þ; ð3; 1Þ; ð3; 5Þ: We may visualize the set A B as the set of six points in the plane with the coordinates that we have just listed. We often draw a diagram (such as Figure 1.1.2) to indicate the Cartesian product of two sets A and B. However, it should be realized that this diagram may be a simplification. For example, if A :¼ fx 2 R : 1 x 2g and B :¼ fy 2 R : 0 y 1 or 2 y 3g, then instead of a rectangle, we should have a drawing such as Figure 1.1.3. Figure 1.1.2 Figure 1.1.3 We will now discuss the fundamental notion of a function or a mapping. To the mathematician of the early nineteenth century, the word ‘‘function’’ meant a definite formula, such as f ðxÞ :¼ x2 þ 3x 5, which associates to each real number x another number f (x). (Here, f (0) ¼ 5, f (1) ¼ 1, f (5) ¼ 35.) This understanding excluded the case of different formulas on different intervals, so that functions could not be defined ‘‘in pieces.’’ As mathematics developed, it became clear that a more general definition of ‘‘function’’ would be useful. It also became evident that it is important to make a clear distinction between the function itself and the values of the function. A revised definition might be: C01 12/08/2010 12:3:17 Page 5 1.1 SETS AND FUNCTIONS 5 A function f from a set A into a set B is a rule of correspondence that assigns to each element x in A a uniquely determined element f (x) in B. But however suggestive this revised definition might be, there is the difficulty of interpreting the phrase ‘‘rule of correspondence.’’ In order to clarify this, we will express the definition entirely in terms of sets; in effect, we will define a function to be its graph. While this has the disadvantage of being somewhat artificial, it has the advantage of being unambiguous and clearer. 1.1.6 Definition Let A and B be sets. Then a function from A to B is a set f of ordered pairs in A B such that for each a 2 A there exists a unique b 2 B with (a, b) 2 f. (In other words, if (a, b) 2 f and (a, b0 ) 2 f, then b ¼ b0 .) The set A of first elements of a function f is called the domain of f and is often denoted by D( f ). The set of all second elements in f is called the range of f and is often denoted by R( f ). Note that, although D( f ) ¼ A, we only have Rðf Þ B. (See Figure 1.1.4.) The essential condition that: ða; bÞ 2 f and ða; b0 Þ 2 f implies that b ¼ b0 is sometimes called the vertical line test. In geometrical terms it says every vertical line x ¼ a with a 2 A intersects the graph of f exactly once. The notation f :A!B is often used to indicate that f is a function from A into B. We will also say that f is a mapping of A into B, or that f maps A into B. If (a, b) is an element in f, it is customary to write b ¼ f ð aÞ or sometimes Figure 1.1.4 a 7! b: A function as a graph If b ¼ f (a), we often refer to b as the value of f at a, or as the image of a under f. Transformations and Machines Aside from using graphs, we can visualize a function as a transformation of the set D( f ) ¼ A into the set Rðf Þ B. In this phraseology, when (a, b) 2 f, we think of f as taking the C01 12/08/2010 12:3:19 6 Page 6 CHAPTER 1 PRELIMINARIES element a from A and ‘‘transforming’’ or ‘‘mapping’’ it into an element b ¼ f (a) in Rðf Þ B. We often draw a diagram, such as Figure 1.1.5, even when the sets A and B are not subsets of the plane. Figure 1.1.5 A function as a transformation There is another way of visualizing a function: namely, as a machine that accepts elements of D( f ) ¼ A as inputs and produces corresponding elements of Rðf Þ B as outputs. If we take an element x 2 D( f ) and put it into f, then out comes the corresponding value f (x). If we put a different element y 2 D( f ) into f, then out comes f (y), which may or may not differ from f (x). If we try to insert something that does not belong to D( f ) into f, we find that it is not accepted, for f can operate only on elements from D( f ). (See Figure 1.1.6.) Figure 1.1.6 A function as a machine This last visualization makes clear the distinction between f and f (x): the first is the machine itself, and the second is the output of the machine f when x is the input. Whereas no one is likely to confuse a meat grinder with ground meat, enough people have confused functions with their values that it is worth distinguishing between them notationally. Direct and Inverse Images Let f : A ! B be a function with domain D( f ) ¼ A and range Rðf Þ B. 1.1.7 Definition If E is a subset of A, then the direct image of E under f is the subset f (E) of B given by f ðEÞ :¼ ff ðxÞ : x 2 Eg: C01 12/08/2010 12:3:20 Page 7 1.1 SETS AND FUNCTIONS 7 If H is a subset of B, then the inverse image of H under f is the subset f 1(H) of A given by f 1 ðHÞ :¼ fx 2 A : f ðxÞ 2 H g: Remark The notation f 1(H) used in this connection has its disadvantages. However, we will use it since it is the standard notation. Thus, if we are given a set E A, then a point y1 2 B is in the direct image f (E) if and only if there exists at least one point x1 2 E such that y1 ¼ f (x1). Similarly, given a set H B, then a point x2 is in the inverse image f 1(H) if and only if y2 :¼ f (x2) belongs to H. (See Figure 1.1.7.) Figure 1.1.7 Direct and inverse images 1.1.8 Examples (a) Let f : R ! R be defined by f (x) :¼ x2. Then the direct image of the set E :¼ fx : 0 x 2g is the set f ðEÞ ¼ fy : 0 y 4g. If G :¼ fy : 0 y 4g, then the inverse image of G is the set f 1 ðGÞ ¼ fx : 2 x 2g. Thus, in this case, we see that f 1 ðf ðEÞÞ 6¼ E. 1 On other the hand, we have f f ðGÞ ¼ G. But if H :¼ fy : 1 y 1g, then we 1 have f f ðH Þ ¼ fy : 0 y 1g 6¼ H. A sketch of the graph of f may help to visualize these sets. (b) Let f : A ! B, and let G, H be subsets of B. We will show that f 1 ðG \ H Þ f 1 ðGÞ \ f 1 ðH Þ: 1 For, if x 2 f ðG \ H Þ, then f ðxÞ 2 G \ H, so that f (x) 2 G and f (x) 2 H. But this implies that x 2 f 1 ðGÞ and x 2 f 1 ðH Þ, whence x 2 f 1 ðGÞ \ f 1 ðH Þ. Thus the stated implication is proved. [The opposite inclusion is also true, so that we actually have set equality between these sets; see Exercise 15.] & Further facts about direct and inverse images are given in the exercises. Special Types of Functions The following definitions identify some very important types of functions. 1.1.9 Definition Let f : A ! B be a function from A to B. (a) The function f is said to be injective (or to be one-one) if whenever x1 6¼ x2, then f ðx1 Þ 6¼ f ðx2 Þ. If f is an injective function, we also say that f is an injection. (b) The function f is said to be surjective (or to map A onto B) if f (A) ¼ B; that is, if the range R( f ) ¼ B. If f is a surjective function, we also say that f is a surjection. C01 12/08/2010 12:3:21 8 Page 8 CHAPTER 1 PRELIMINARIES (c) If f is both injective and surjective, then f is said to be bijective. If f is bijective, we also say that f is a bijection. In order to prove that a function f is injective, we must establish that: for all x1 ; x2 in A; if f ðx1 Þ ¼ f ðx2 Þ; then x1 ¼ x2 : To do this we assume that f (x1) ¼ f (x2) and show that x1 ¼ x2. [In other words, the graph of f satisfies the first horizontal line test: Every horizontal line y ¼ b with b 2 B intersects the graph f in at most one point.] To prove that a function f is surjective, we must show that for any b 2 B there exists at least one x 2 A such that f (x) ¼ b. [In other words, the graph of f satisfies the second horizontal line test: Every horizontal line y ¼ b with b 2 B intersects the graph f in at least one point.] 1.1.10 Example Let A :¼ fx 2 R : x 6¼ 1g and define f ðxÞ :¼ 2x=ðx 1Þ for all x 2 A. To show that f is injective, we take x1 and x2 in A and assume that f (x1) ¼ f (x2). Thus we have 2x1 2x2 ¼ ; x1 1 x2 1 which implies that x1 ðx2 1Þ ¼ x2 ðx1 1Þ, and hence x1 ¼ x2. Therefore f is injective. To determine the range of f, we solve the equation y ¼ 2x=ðx 1Þ for x in terms of y. We obtain x ¼ y=ðy 2Þ, which is meaningful for y 6¼ 2. Thus the range of f is the set & B :¼ fy 2 R : y 6¼ 2g. Thus, f is a bijection of A onto B. Inverse Functions If f is a function from A into B, then f is a special subset of A B (namely, one passing the vertical line test.) The set of ordered pairs in B A obtained by interchanging the members of ordered pairs in f is not generally a function. (That is, the set f may not pass both of the horizontal line tests.) However, if f is a bijection, then this interchange does lead to a function, called the ‘‘inverse function’’ of f. 1.1.11 Definition If f : A ! B is a bijection of A onto B, then g :¼ fðb; aÞ 2 B A : ða; bÞ 2 f g is a function on B into A. This function is called the inverse function of f, and is denoted by f 1. The function f 1 is also called the inverse of f. We can also express the connection between f and its inverse f 1 by noting that D( f ) ¼ R( f 1) and R( f ) ¼ D( f 1) and that b ¼ f ðaÞ if and only if a¼f 1 ðbÞ : For example, we saw in Example 1.1.10 that the function f ðxÞ :¼ 2x x1 is a bijection of A :¼ fx 2 R : x 6¼ 1g onto the set B :¼ fy 2 R : y 6¼ 2g. Solving y ¼ f(x) for x in terms of y, we find the function inverse to f is given by f 1 ðyÞ :¼ y for y 2 B: y2 C01 12/08/2010 12:3:22 Page 9 1.1 SETS AND FUNCTIONS 9 Remark We introduced the notation f 1(H) in Definition 1.1.7. It makes sense even if f does not have an inverse function. However, if the inverse function f 1 does exist, then f 1(H) is the direct image of the set H B under f 1. Composition of Functions It often happens that we want to ‘‘compose’’ two functions f, g by first finding f (x) and then applying g to get g ( f (x)); however, this is possible only when f(x) belongs to the domain of g. In order to be able to do this for all f (x), we must assume that the range of f is contained in the domain of g. (See Figure 1.1.8.) Figure 1.1.8 The composition of f and g 1.1.12 Definition If f : A ! B and g : B ! C, and if Rðf Þ DðgÞ ¼ B, then the composite function g f (note the order!) is the function from A into C defined by ðg f ÞðxÞ :¼ gðf ðxÞÞ for all x 2 A: 1.1.13 Examples (a) The order of the composition must be carefully noted. For, let f and g be the functions whose values at x 2 R are given by f ðxÞ :¼ 2x and gðxÞ :¼ 3x2 1: Since DðgÞ ¼ R and Rðf Þ R ¼ DðgÞ, then the domain D(g f ) is also equal to R, and the composite function g f is given by ðg f ÞðxÞ ¼ 3ð2xÞ2 1 ¼ 12x2 1: On the other hand, the domain of the composite function f g is also R, but ð f gÞðxÞ ¼ 2 3x2 1 ¼ 6x2 2: Thus, in this case, we have g f 6¼ f g. (b) In considering g f, some care must be exercised to be sure that the range of f is contained in the domain of g. For example, if pffiffiffi and gðxÞ :¼ x ; f ðxÞ :¼ 1 x2 then, since DðgÞ ¼ fx : x 0g, the composite function g pffiffiffiffiffiffiffiffiffiffiffiffiffi ðg f ÞðxÞ ¼ 1 x2 only for x 2 Dðf Þ that satisfy f (x) f is given by the formula 0; that is, for x satisfying 1 x 1. C01 12/08/2010 12:3:22 10 Page 10 CHAPTER 1 PRELIMINARIES We note that if we reverse the order, then the composition f g is given by the formula ðf gÞðxÞ ¼ 1 x; but only for those x in the domain DðgÞ ¼ fx : x 0g. & We now give the relationship between composite functions and inverse images. The proof is left as an instructive exercise. 1.1.14 Theorem Let f : A ! B and g : B ! C be functions and let H be a subset of C. Then we have ðg f Þ1 ðH Þ ¼ f 1 g1 ðH Þ : Note the reversal in the order of the functions. Restrictions of Functions If f : A ! B is a function and if A1 A, we can define a function f1: A1 ! B by f 1 ðxÞ :¼ f ðxÞ for x 2 A1 : The function f1 is called the restriction of f to A1. Sometimes it is denoted by f 1 ¼ f jA1. It may seem strange to the reader that one would ever choose to throw away a part of a function, but there are some good reasons for doing so. For example, if f : R ! R is the squaring function: f ðxÞ :¼ x2 for x 2 R; then f is not injective, so it cannot have an inverse function. However, if we restrict f to the set A1 :¼ fx : x 0g, then the restriction f jA1 is a bijection of A1 onto A1. Therefore, this restriction has an inverse function, which is the positive square root function. (Sketch a graph.) Similarly, the trigonometric functions S(x) :¼ sin x and C(x) :¼ cos x are not injective on all of R. However, by making suitable restrictions of these functions, one can obtain the inverse sine and the inverse cosine functions that the reader has undoubtedly already encountered. Exercises for Section 1.1 1. Let A :¼ fk : k 2 N; k 20g; B :¼ f3k 1 : k 2 N g; and C :¼ f2k þ 1 : k 2 N g: Determine the sets: (a) A \ B \ C, (b) ðA \ BÞnC, (c) ðA \ C ÞnB. 2. Draw diagrams to simplify and identify the following sets: (a) An(BnA), (b) An(AnB), (c) A \ ðBnAÞ. 3. If A and B are sets, show that A B if and only if A \ B ¼ A. 4. Prove the second De Morgan Law [Theorem 1.1.4(b)]. 5. Prove the Distributive Laws: (a) A \ ðB [ CÞ ¼ ðA \ BÞ [ ðA \ CÞ, (b) A [ ðB \ CÞ ¼ ðA [ BÞ \ ðA [ CÞ. C01 12/08/2010 12:3:23 Page 11 11 1.1 SETS AND FUNCTIONS 6. The symmetric difference of two sets A and B is the set D of all elements that belong to either A or B but not both. Represent D with a diagram. (a) Show that D ¼ ðAnBÞ [ ðBnAÞ. (b) Show that D is also given by D ¼ ðA [ BÞnðA \ BÞ. 7. For each n 2 N, let An ¼ fðn þ 1Þk : k 2 N g. (a) What is A1 \ A2 ? (b) Determine the sets [fAn : n 2 N g and \fAn : n 2 N g. 8. Draw diagrams in the plane of the Cartesian products A B for the given sets A and B. (a) A ¼ fx 2 R : 1 x 2 or 3 x 4g; B ¼ fx 2 R : x ¼ 1 or x ¼ 2g. (b) A ¼ f1; 2; 3g; B ¼ fx 2 R : 1 x 3g. 9. Let A :¼ B :¼ fx 2 R : 1 x 1g and consider the subset C :¼ ðx; yÞ : x2 þ y2 ¼ 1 of A B. Is this set a function? Explain. 10. Let f ðxÞ :¼ 1=x2 ; x 6¼ 0; x 2 R. (a) Determine the direct image f (E) where E :¼ fx 2 R : 1 x 2g. (b) Determine the inverse image f 1(G) where G :¼ fx 2 R : 1 x 4g. 11. Let g(x) :¼ x2 and f (x) :¼ x þ 2 for x 2 R, and let h be the composite function h :¼ g (a) Find the direct image h(E) of E :¼ fx 2 R : 0 x 1g. (b) Find the inverse image h1(G) of G :¼ fx 2 R : 0 x 4g. f. 12. Let f (x) :¼ x2 for x 2 R, and let E :¼ fx 2 R : 1 x 0g and F :¼ fx 2 R : 0 x 1g. Show that E \ F ¼ f0g and f ð E \ F Þ ¼ f0g, while f ðEÞ ¼ f ðF Þ ¼ fy 2 R : 0 y 1g. Hence f ð E \ F Þ is a proper subset of f ðEÞ \ f ðF Þ. What happens if 0 is deleted from the sets E and F? 13. Let f and E, F be as in Exercise 12. Find the sets EnF and f (E)n f ( f ) and show that it is not true that f ð EnF Þ f ðEÞn f ðF Þ. 14. Show that if f : A ! B and E, F are subsets of A, then f ð E [ F Þ ¼ f ðEÞ [ f ðF Þ and f ð E \ F Þ f ðEÞ \ f ðF Þ. 15. Show that if f : A ! B and G, H are subsets of B, then f 1 ðG [ H Þ ¼ f 1 ðGÞ [ f 1 ðH Þ and f 1 ðG \ H Þ ¼ f 1 ðGÞ \ f 1 ðH Þ. pffiffiffiffiffiffiffiffiffiffiffiffiffi 16. Show that the function f defined by f ðxÞ :¼ x= x2 þ 1; x 2 R , is a bijection of R onto fy : 1 < y < 1g. 17. For a, b 2 R with a < b, find an explicit bijection of A :¼ fx : a < x < bg onto B :¼ fy : 0 < y < 1g. 18. (a) Give an example of two functions f, g on R to R such that f 6¼ g, but such that f g ¼ g f. (b) Give an example of three functions f, g, h on R such that f ðg þ hÞ 6¼ f g þ f h. 19. (a) Show that if f : A ! B is injective and E A, then f 1 ð f ðEÞÞ ¼ E. Give an example to show that equality need not hold if f is not injective. (b) Show that if f : A ! B is surjective and H B, then f f 1 ðHÞ ¼ H. Give an example to show that equality need not hold if f is not surjective. 20. (a) Suppose that f is an injection. Show that f 1 f (x) ¼ x for all x 2 D( f ) and that f f 1(y) ¼ y for all y 2 R( f ). (b) If f is a bijection of A onto B, show that f 1 is a bijection of B onto A. 21. Prove that if f : A ! B is bijective and g : B ! C is bijective, then the composite g bijective map of A onto C. f is a 22. Let f : A ! B and g : B ! C be functions. (a) Show that if g f is injective, then f is injective. (b) Show that if g f is surjective, then g is surjective. 23. Prove Theorem 1.1.14. 24. Let f, g be functions such that (g f )(x) ¼ x for all x 2 D( f ) and ( f g)(y) ¼ y for all y 2 D (g). Prove that g ¼ f 1 . C01 12/08/2010 12:3:23 12 Page 12 CHAPTER 1 PRELIMINARIES Section 1.2 Mathematical Induction Mathematical Induction is a powerful method of proof that is frequently used to establish the validity of statements that are given in terms of the natural numbers. Although its utility is restricted to this rather special context, Mathematical Induction is an indispensable tool in all branches of mathematics. Since many induction proofs follow the same formal lines of argument, we will often state only that a result follows from Mathematical Induction and leave it to the reader to provide the necessary details. In this section, we will state the principle and give several examples to illustrate how inductive proofs proceed. We shall assume familiarity with the set of natural numbers: N ¼ f1; 2; 3; . . .g; with the usual arithmetic operations of addition and multiplication, and with the meaning of a natural number being less than another one. We will also assume the following fundamental property of N. 1.2.1 Well-Ordering Property of N Every nonempty subset of N has a least element. A more detailed statement of this property is as follows: If S is a subset of N and if S 6¼ ;, then there exists m 2 S such that m k for all k 2 S. On the basis of the Well-Ordering Property, we shall derive a version of the Principle of Mathematical Induction that is expressed in terms of subsets of N. 1.2.2 Principle of Mathematical Induction Let S be a subset of N that possesses the two properties: (1) The number 1 2 S. (2) For every k 2 N, if k 2 S, then k þ 1 2 S. Then we have S ¼ N. Proof. Suppose to the contrary that S 6¼ N. Then the set NnS is not empty, so by the WellOrdering Principle it has a least element m. Since 1 2 S by hypothesis (1), we know that m > 1. But this implies that m 1 is also a natural number. Since m 1 < m and since m is the least element in N such that m 2 = S, we conclude that m 1 2 S. We now apply hypothesis (2) to the element k :¼ m 1 in S, to infer that k þ 1 ¼ ðm 1Þ þ 1 ¼ m belongs to S. But this statement contradicts the fact that m 2 = S. Since m was obtained from the assumption that NnS is not empty, we have obtained a contradiction. Therefore we must have S ¼ N. Q.E.D. The Principle of Mathematical Induction is often set forth in the framework of statements about natural numbers. If P(n) is a meaningful statement about n 2 N, then P(n) may be true for some values of n and false for others. For example, if P1(n) is the statement: ‘‘n2 ¼ n,’’ then P1(1) is true while P1(n) is false for all n > 1, n 2 N. On the other hand, if P2(n) is the statement: ‘‘n2 > 1,’’ then P2(1) is false, while P2(n) is true for all n > 1. In this context, the Principle of Mathematical Induction can be formulated as follows. For each n 2 N , let P(n) be a statement about n. Suppose that: (10 ) P(1) is true. (20 ) For every k 2 N , if P(k) is true, then P(k þ 1) is true. Then P(n) is true for all n 2 N. C01 12/08/2010 12:3:24 Page 13 1.2 MATHEMATICAL INDUCTION 13 The connection with the preceding version of Mathematical Induction, given in 1.2.2, is made by letting S :¼ fn 2 N : PðnÞ is trueg. Then the conditions (1) and (2) of 1.2.2 correspond exactly to the conditions (10 ) and (20 ), respectively. The conclusion that S ¼ N in 1.2.2 corresponds to the conclusion that P(n) is true for all n 2 N. In (20 ) the assumption ‘‘if P(k) is true’’ is called the induction hypothesis. In establishing (20 ), we are not concerned with the actual truth or falsity of P(k), but only with the validity of the implication ‘‘if P(k), then P(k þ 1).’’ For example, if we consider the statements P(n): ‘‘n ¼ n þ 5,’’ then (20 ) is logically correct, for we can simply add 1 to both sides of P(k) to obtain P(k þ 1). However, since the statement P(1): ‘‘1 ¼ 6’’ is false, we cannot use Mathematical Induction to conclude that n ¼ n þ 5 for all n 2 N. It may happen that statements P(n) are false for certain natural numbers but then are true for all n n0 for some particular n0. The Principle of Mathematical Induction can be modified to deal with this situation. We will formulate the modified principle, but leave its verification as an exercise. (See Exercise 12.) 1.2.3 Principle of Mathematical Induction (second version) Let n0 2 N and let P(n) be a statement for each natural number n n0. Suppose that: (1) The statement P(n0) is true. (2) For all k n0, the truth of P(k) implies the truth of P(k þ 1). Then P(n) is true for all n n 0. Sometimes the number n0 in (1) is called the base, since it serves as the starting point, and the implication in (2), which can be written PðkÞ ) Pðk þ 1Þ, is called the bridge, since it connects the case k to the case k þ 1. The following examples illustrate how Mathematical Induction is used to prove assertions about natural numbers. 1.2.4 Examples (a) For each n 2 N, the sum of the first n natural numbers is given by 1þ2þ þ n ¼ 12 nðn þ 1Þ: To prove this formula, we let S be the set of all n 2 N for which the formula is true. We must verify that conditions (1) and (2) of 1.2.2 are satisfied. If n ¼ 1, then we have 1 ¼ 12 1 ð1 þ 1Þ so that 1 2 S, and (1) is satisfied. Next, we assume that k 2 S and wish to infer from this assumption that k þ 1 2 S. Indeed, if k 2 S, then 1þ2þ þ k ¼ 12 kðk þ 1Þ: If we add k þ 1 to both sides of the assumed equality, we obtain 1þ2þ þ k þ ðk þ 1Þ¼ 12 kðk þ 1Þ þ ðk þ 1Þ ¼ 12 ðk þ 1Þðk þ 2Þ: Since this is the stated formula for n ¼ k þ 1, we conclude that k þ 1 2 S. Therefore, condition (2) of 1.2.2 is satisfied. Consequently, by the Principle of Mathematical Induction, we infer that S ¼ N, so the formula holds for all n 2 N. (b) For each n 2 N, the sum of the squares of the first n natural numbers is given by 12 þ 22 þ þ n2 ¼ 16 nðn þ 1Þð2n þ 1Þ: C01 12/08/2010 12:3:24 14 Page 14 CHAPTER 1 PRELIMINARIES To establish this formula, we note that it is true for n ¼ 1, since 12 ¼ 16 1 2 3. If we assume it is true for k, then adding (k þ 1)2 to both sides of the assumed formula gives 12 þ 22 þ þ k2 þ ðk þ 1Þ2 ¼ 16 kðk þ 1Þð2k þ 1Þ þ ðk þ 1Þ2 ¼ 16 ðk þ 1Þ 2k2 þ k þ 6k þ 6 ¼ 16 ðk þ 1Þðk þ 2Þð2k þ 3Þ: Consequently, the formula is valid for all n 2 N. (c) Given two real numbers a and b, we will prove that a b is a factor of an bn for all n 2 N. First we see that the statement is clearly true for n ¼ 1. If we now assume that a b is a factor of ak bk, then akþ1 bkþ1 ¼ akþ1 abk þ abk bkþ1 ¼ a ak bk þ bk ða bÞ: By the induction hypothesis, a b is a factor of a (ak bk) and it is plainly a factor of bk(a b). Therefore, a b is a factor of akþ1 bkþl, and it follows from Mathematical Induction that a b is a factor of an bn for all n 2 N. A variety of divisibility results can be derived from this fact. For example, since 11 7 ¼ 4, we see that 11n 7n is divisible by 4 for all n 2 N. (d) The inequality 2n > 2n þ 1 is false for n ¼ 1, 2, but it is true for n ¼ 3. If we assume that 2k > 2k þ 1, then multiplication by 2 gives, when 2k þ 2 > 3, the inequality 2kþ1 > 2ð2k þ 1Þ ¼ 4k þ 2 ¼ 2k þ ð2k þ 2Þ > 2k þ 3 ¼ 2ðk þ 1Þ þ 1: Since 2k þ 2 > 3 for all k 1, the bridge is valid for all k 1 (even though the statement is false for k ¼ 1, 2). Hence, with the base n0 ¼ 3, we can apply Mathematical Induction to conclude that the inequality holds for all n 3. (e) The inequality 2n (n þ 1)! can be established by Mathematical Induction. We first observe that it is true for n ¼ 1, since 21 ¼ 2 ¼ 1 þ 1. If we assume that k 2 (k þ 1)!, it follows from the fact that 2 k þ 2 that 2kþ1 ¼ 2 2k 2ðk þ 1Þ! ðk þ 2Þðk þ 1Þ! ¼ ðk þ 2Þ!: Thus, if the inequality holds for k, then it also holds for k þ 1. Therefore, Mathematical Induction implies that the inequality is true for all n 2 N. (f) If r 2 R, r 6¼ 1, and n 2 N, then 1 þ r þ r2 þ þ rn ¼ 1 rnþ1 : 1r This is the formula for the sum of the terms in a ‘‘geometric progression.’’ It can be established using Mathematical Induction as follows. First, if n ¼ 1, then 1 þ r ¼ (1 r2)=(1 r). If we assume the truth of the formula for n ¼ k and add the term rkþ1 to both sides, we get (after a little algebra) 1 þ r þ rk þ þ rkþ1 ¼ 1 rkþ1 1 rkþ2 þ rkþ1 ¼ ; 1r 1r which is the formula for n ¼ k þ 1. Therefore, Mathematical Induction implies the validity of the formula for all n 2 N. C01 12/08/2010 12:3:25 Page 15 1.2 MATHEMATICAL INDUCTION 15 [This result can also be proved without using Mathematical Induction. If we let sn :¼ 1 þ r þ r2 þ þ rn , then rsn ¼ r þ r2 þ þ rnþ1 , so that ð1 rÞsn ¼ sn rsn ¼ 1 rnþ1 : If we divide by 1 r, we obtain the stated formula.] (g) Careless use of the Principle of Mathematical Induction can lead to obviously absurd conclusions. The reader is invited to find the error in the ‘‘proof’’ of the following assertion. Claim: If n 2 N and if the maximum of the natural numbers p and q is n, then p ¼ q. ‘‘Proof.’’ Let S be the subset of N for which the claim is true. Evidently, 1 2 S since if p, q 2 N and their maximum is 1, then both equal 1 and so p ¼ q. Now assume that k 2 S and that the maximum of p and q is k þ 1. Then the maximum of p 1 and q 1 is k. But since k 2 S, then p 1 ¼ q 1 and therefore p ¼ q. Thus, k þ 1 2 S, and we conclude that the assertion is true for all n 2 N. (h) There are statements that are true for many natural numbers but that are not true for all of them. For example, the formula p (n) :¼ n2 n þ 41 gives a prime number for n ¼ 1, 2, . . . , 40. However, p(41) is obviously divisible by 41, so it is not a prime number. Another version of the Principle of Mathematical Induction is sometimes quite useful. It is called the ‘‘Principle of Strong Induction,’’ even though it is in fact equivalent to 1.2.2. 1.2.5 Principle of Strong Induction Let S be a subset of N such that (100 ) 1 2 S. (200 ) For every k 2 N, if f1; 2; . . . ; kg S, then k þ 1 2 S. Then S ¼ N. We will leave it to the reader to establish the equivalence of 1.2.2 and 1.2.5. Exercises for Section 1.2 1. Prove that 1=1 2 þ 1=2 3 þ þ 1=nðn þ 1Þ ¼ n=ðn þ 1Þ for all n 2 N. 1 2 þ n ¼ 2 nðn þ 1Þ for all n 2 N. 2. Prove that 1 þ 2 þ 3 3 3 3. Prove that 3 þ 11 þ þ ð8n 5Þ ¼ 4n2 n for all n 2 N. þ ð2n 1Þ2 ¼ ð4n3 nÞ=3 for all n 2 N. 4. Prove that 12 þ 32 þ 5. Prove that 1 2 þ 3 þ 2 2 2 þ ð1Þnþ1 n2 ¼ ð1Þnþ1 nðn þ 1Þ=2 for all n 2 N. 6. Prove that n3 þ 5n is divisible by 6 for all n 2 N. 7. Prove that 52n 1 is divisible by 8 for all n 2 N. 8. Prove that 5n 4n 1 is divisible by 16 for all n 2 N. 9. Prove that n3 þ (n þ 1)3 þ (n þ 2)3 is divisible by 9 for all n 2 N. 10. Conjecture a formula for the sum 1=1 3 þ 1=3 5 þ conjecture by using Mathematical Induction. þ 1=ð2n 1Þð2n þ 1Þ, and prove your 11. Conjecture a formula for the sum of the first n odd natural numbers 1 þ 3 þ prove your formula by using Mathematical Induction. 12. Prove the Principle of Mathematical Induction 1.2.3 (second version). þ ð2n 1Þ, and C01 12/08/2010 12:3:25 16 Page 16 CHAPTER 1 PRELIMINARIES 13. Prove that n < 2n for all n 2 N. 14. Prove that 2n < n! for all n n2 15. Prove that 2n 3 2 4, n 2 N. for all n 5, n 2 N. 16. Find all natural numbers n such that n2 < 2n. Prove your assertion. 17. Find the largest natural number m such that n3 n is divisible by m for all n 2 N. Prove your assertion. pffiffiffi pffiffiffi pffiffiffi pffiffiffi 18. Prove that 1= 1 þ 1= 2 þ þ 1= n > n for all n 2 N, n > 1. 19. Let S be a subset of N such that (a) 2k 2 S for all k 2 N, and (b) if k 2 S and k k 1 2 S. Prove that S ¼ N. 2, then 20. Let the numbers xn be defined as follows: x1 :¼ 1, x2 :¼ 2, and xnþ2 :¼ 12 ðxnþ1 þ xn Þ for all n 2 N. Use the Principle of Strong Induction (1.2.5) to show that 1 xn 2 for all n 2 N. Section 1.3 Finite and Infinite Sets When we count the elements in a set, we say ‘‘one, two, three, . . . ,’’ stopping when we have exhausted the set. From a mathematical perspective, what we are doing is defining a bijective mapping between the set and a portion of the set of natural numbers. If the set is such that the counting does not terminate, such as the set of natural numbers itself, then we describe the set as being infinite. The notions of ‘‘finite’’ and ‘‘infinite’’ are extremely primitive, and it is very likely that the reader has never examined these notions very carefully. In this section we will define these terms precisely and establish a few basic results and state some other important results that seem obvious but whose proofs are a bit tricky. These proofs can be found in Appendix B and can be read later. 1.3.1 Definition (a) The empty set ; is said to have 0 elements. (b) If n 2 N, a set S is said to have n elements if there exists a bijection from the set N n :¼ f1; 2; . . . ; ng onto S. (c) A set S is said to be finite if it is either empty or it has n elements for some n 2 N. (d) A set S is said to be infinite if it is not finite. Since the inverse of a bijection is a bijection, it is easy to see that a set S has n elements if and only if there is a bijection from S onto the set {1, 2, . . . , n}. Also, since the composition of two bijections is a bijection, we see that a set S1 has n elements if and only if there is a bijection from S1 onto another set S2 that has n elements. Further, a set T1 is finite if and only if there is a bijection from T1 onto another set T2 that is finite. It is now necessary to establish some basic properties of finite sets to be sure that the definitions do not lead to conclusions that conflict with our experience of counting. From the definitions, it is not entirely clear that a finite set might not have n elements for more than one value of n. Also it is conceivably possible that the set N :¼ f1; 2; 3; . . .g might be a finite set according to this definition. The reader will be relieved that these possibilities do not occur, as the next two theorems state. The proofs of these assertions, which use the fundamental properties of N described in Section 1.2, are given in Appendix B. 1.3.2 Uniqueness Theorem If S is a finite set, then the number of elements in S is a unique number in N. C01 12/08/2010 12:3:25 Page 17 1.3 FINITE AND INFINITE SETS 17 1.3.3 Theorem The set N of natural numbers is an infinite set. The next result gives some elementary properties of finite and infinite sets. 1.3.4 Theorem (a) If A is a set with m elements and B is a set with n elements and if A \ B ¼ ; , then A [ B has m þ n elements. (b) If A is a set with m 2 N elements and C A is a set with 1 element, then AnC is a set with m 1 elements. (c) If C is an infinite set and B is a finite set, then CnB is an infinite set. Proof. (a) Let f be a bijection of N m onto A, and let g be a bijection of N n onto B. We define h on N mþn by h(i) :¼ f (i) for i ¼ 1; . . . ; m and h(i) :¼ g(i m) for i ¼ m þ 1; . . . ; m þ n. We leave it as an exercise to show that h is a bijection from N mþn onto A [ B. The proofs of parts (b) and (c) are left to the reader, see Exercise 2. Q.E.D. It may seem ‘‘obvious’’ that a subset of a finite set is also finite, but the assertion must be deduced from the definitions. This and the corresponding statement for infinite sets are established next. 1.3.5 Theorem Suppose that S and T are sets and that T S. (a) If S is a finite set, then T is a finite set. (b) If T is an infinite set, then S is an infinite set. Proof. (a) If T ¼ ;, we already know that T is a finite set. Thus we may suppose that T 6¼ ;. The proof is by induction on the number of elements in S. If S has 1 element, then the only nonempty subset T of S must coincide with S, so T is a finite set. Suppose that every nonempty subset of a set with k elements is finite. Now let S be a set having k þ 1 elements (so there exists a bijection f of N kþ1 onto S), and let T S. If f ð k þ 1Þ 2 = T, we can consider T to be a subset of S1 :¼ Snff ð k þ 1Þg, which has k elements by Theorem 1.3.4(b). Hence, by the induction hypothesis, T is a finite set. On the other hand, if f ð k þ 1Þ 2 T, then T 1 :¼ Tnff ð k þ 1Þg is a subset of S1. Since S1 has k elements, the induction hypothesis implies that T1 is a finite set. But this implies that T ¼ T 1 [ ff ð k þ 1Þg is also a finite set. (b) This assertion is the contrapositive of the assertion in (a). (See Appendix A for a discussion of the contrapositive.) Q.E.D. Countable Sets We now introduce an important type of infinite set. 1.3.6 Definition (a) A set S is said to be denumerable (or countably infinite) if there exists a bijection of N onto S. (b) A set S is said to be countable if it is either finite or denumerable. (c) A set S is said to be uncountable if it is not countable. From the properties of bijections, it is clear that S is denumerable if and only if there exists a bijection of S onto N. Also a set S1 is denumerable if and only if there exists a C01 12/08/2010 12:3:25 18 Page 18 CHAPTER 1 PRELIMINARIES bijection from S1 onto a set S2 that is denumerable. Further, a set T1 is countable if and only if there exists a bijection from T1 onto a set T2 that is countable. Finally, an infinite countable set is denumerable. 1.3.7 Examples (a) The set E :¼ f2n : n 2 N g of even natural numbers is denumerable, since the mapping f : N ! E defined by f (n) :¼ 2n for n 2 N is a bijection of N onto E. Similarly, the set O :¼ f2n 1 : n 2 N g of odd natural numbers is denumerable. (b) The set Z of all integers is denumerable. To construct a bijection of N onto Z, we map 1 onto 0, we map the set of even natural numbers onto the set N of positive integers, and we map the set of odd natural numbers onto the negative integers. This mapping can be displayed by the enumeration: Z ¼ f0; 1; 1; 2; 2; 3; 3; . . .g: (c) The union of two disjoint denumerable sets is denumerable. Indeed, if A ¼ fa1 ; a2 ; a3 ; . . .g and B ¼ fb1 ; b2 ; b3 ; . . .g, we can enumerate the elements of A [ B as: & a 1 ; b1 ; a2 ; b2 ; a3 ; b3 ; . . . : 1.3.8 Theorem The set N N is denumerable. Informal Proof. Recall that N N consists of all ordered pairs (m, n), where m, n 2 N. We can enumerate these pairs as: ð1; 1Þ; ð1; 2Þ; ð2; 1Þ; ð1; 3Þ; ð2; 2Þ; ð3; 1Þ; ð1; 4Þ; . . . ; according to increasing sum m þ n, and increasing m. (See Figure 1.3.1.) Q.E.D. The enumeration just described is an instance of a ‘‘diagonal procedure,’’ since we move along diagonals that each contain finitely many terms as illustrated in Figure 1.3.1. The bijection indicated by the diagram can be derived as follows. We first notice that the first diagonal has one point, the second diagonal has two points, and so on, with k points in the kth diagonal. Applying the formula in Example 1.2.4(a), we see that the total number of points in diagonals 1 through k is given by cð k Þ ¼ 1 þ 2 þ Figure 1.3.1 þ k ¼ 12 kðk þ 1Þ The set N N C01 12/08/2010 12:3:26 Page 19 1.3 FINITE AND INFINITE SETS 19 The point (m, n) lies in the kth diagonal when k ¼ m þ n 1, and it is the mth point in that diagonal as we move downward from left to right. (For example, the point (3, 2) lies in the 4th diagonal since 3 þ 2 1 ¼ 4, and it is the 3rd point in that diagonal.) Therefore, in the counting scheme displayed by Figure 1.3.1, we count the point (m, n) by first counting the points in the first k 1 ¼ m þ n 2 diagonals and then adding m. Therefore, the counting function h : N N ! N is given by hðm; nÞ :¼ cðm þ n 2Þ þ m ¼ 12 ðm þ n 2Þðm þ n 1Þ þ m: For example, the point (3, 2) is counted as number hð3; 2Þ ¼ 12 3 4 þ 3 ¼ 9, as shown by Figure 1.3.1. Similarly, the point (17, 25) is counted as number h (17, 25) ¼ c(40) þ 17 ¼ 837. This geometric argument leading to the counting formula has been suggestive and convincing, but it remains to be proved that h is, in fact, a bijection of N N onto N. A detailed proof is given in Appendix B. The construction of an explicit bijection between sets is often complicated. The next two results are useful in establishing the countability of sets, since they do not involve showing that certain mappings are bijections. The first result may seem intuitively clear, but its proof is rather technical; it will be given in Appendix B. 1.3.9 Theorem Suppose that S and T are sets and that T S. (a) If S is a countable set, then T is a countable set. (b) If T is an uncountable set, then S is an uncountable set. 1.3.10 Theorem The following statements are equivalent: (a) S is a countable set. (b) There exists a surjection of N onto S. (c) There exists an injection of S into N. Proof. (a) ) (b) define H on N by If S is finite, there exists a bijection h of some set N n onto S and we HðkÞ :¼ hðkÞ for hðnÞ for k ¼ 1; . . . ; n; k > n: Then H is a surjection of N onto S. If S is denumerable, there exists a bijection H of N onto S, which is also a surjection of N onto S. (b) ) (c) If H is a surjection of N onto S, we define H 1 : S ! N by letting H1(s) be the least element in the set H 1 ðsÞ :¼ fn 2 N : HðnÞ ¼ sg. To see that H1 is an injection of S into N, note that if s, t 2 S and nst :¼ H 1 ðsÞ ¼ H 1 ðtÞ, then s ¼ H(nst) ¼ t. (c) ) (a) If H1 is an injection of S into N, then it is a bijection of S onto H 1 ðSÞ N. By Theorem 1.3.9(a), H1(S) is countable, whence the set S is countable. Q.E.D. 1.3.11 Theorem The set Q of all rational numbers is denumerable. Proof. The idea of the proof is to observe that the set Q þ of positive rational numbers is contained in the enumeration: 1 1 2 1 2 3 1 1; 2; 1; 3; 2; 1; 4;...; C01 12/08/2010 12:3:26 20 Page 20 CHAPTER 1 PRELIMINARIES which is another ‘‘diagonal mapping’’ (see Figure 1.3.2). However, this mapping is not an injection, since the different fractions 12 and 24 represent the same rational number. Figure 1.3.2 The set Q þ To proceed more formally, note that since N N is countable (by Theorem 1.3.8), it follows from Theorem 1.3.10(b) that there exists a surjection f of N onto N N. If g : N N ! Q þ is the mapping that sends the ordered pair (m, n) into the rational number having a representation m=n, then g is a surjection onto Q þ . Therefore, the composition g f is a surjection of N onto Q þ , and Theorem 1.3.10 implies that Q þ is a countable set. Similarly, the set Q of all negative rational numbers is countable. It follows as in Example 1.3.7(b) that the set Q ¼ Q [ f0g [ Q þ is countable. Since Q contains N, it must be a denumerable set. Q.E.D. The next result is concerned with unions of sets. In view of Theorem 1.3.10, we need not be worried about possible overlapping of the sets. Also, we do not have to construct a bijection. 1.3.12 Theorem If Am is a countable set for each m 2 N, then the union A :¼ is countable. S1 m¼1 Am Proof. For each m 2 N, let wm be a surjection of N onto Am. We define b : N N ! A by bðm; nÞ :¼ wm ðnÞ: We claim that b is a surjection. Indeed, if a 2 A, then there exists a least m 2 N such that a 2 Am, whence there exists a least n 2 N such that a ¼ wm ðnÞ. Therefore, a ¼ b (m, n). Since N N is countable, it follows from Theorem 1.3.10 that there exists a surjection f : N ! N N whence b f is a surjection of N onto A. Now apply Theorem 1.3.10 again to conclude that A is countable. Q.E.D. Remark A less formal (but more intuitive) way to see the truth of Theorem 1.3.12 is to enumerate the elements of Am, m 2 N, as: A1 ¼ fa11 ; a12 ; a13 ; . . .g; A2 ¼ fa21 ; a22 ; a23 ; . . .g; A3 ¼ fa31 ; a32 ; a33 ; . . .g; : C01 12/08/2010 12:3:26 Page 21 1.3 FINITE AND INFINITE SETS 21 We then enumerate this array using the ‘‘diagonal procedure’’: a11 ; a12 ; a21 ; a13 ; a22 ; a31 ; a14 ; . . . ; as was displayed in Figure 1.3.1. Georg Cantor Georg Cantor (1845–1918) was born in St. Petersburg, Russia. His father, a Danish businessman working in Russia, moved the family to Germany several years later. Cantor studied briefly at Zurich, then went to the University of Berlin, the best in mathematics at the time. He received his doctorate in 1869, and accepted a position at the University of Halle, where he worked alone on his research, but would occasionally travel the seventy miles to Berlin to visit colleagues. Cantor is known as the founder of modern set theory and he was the first to study the concept of infinite set in rigorous detail. In 1874 he proved that Q is countable and, in contrast, that R is uncountable (see Section 2.5), exhibiting two kinds of infinity. In a series of papers he developed a general theory of infinite sets, including some surprising results. In 1877 he proved that the two-dimensional unit square in the plane could be put into one-one correspondence with the unit interval on the line, a result he sent in a letter to his colleague Richard Dedekind in Berlin, writing ‘‘I see it, but I do not believe it.’’ Cantor’s Theorem on sets of subsets shows there are many different orders of infinity and this led him to create a theory of ‘‘transfinite’’ numbers that he published in 1895 and 1897. His work generated considerable controversy among mathematicians of that era, but in 1904, London’s Royal Society awarded Cantor the Sylvester Medal, its highest honor. Beginning in 1884, he suffered from episodes of depression that increased in severity as the years passed. He was hospitalized several times for nervous breakdowns in the Halle Nervenklinik and spent the last seven months of his life there. We close this section with one of Cantor’s more remarkable theorems. 1.3.13 Cantor’s Theorem If A is any set, then there is no surjection of A onto the set PðAÞ of all subsets of A. Proof. Suppose that w : A ! PðAÞ is a surjection. Since w(a) is a subset of A, either a belongs to w(a) or it does not belong to this set. We let D :¼ fa 2 A : a 2 = wðaÞg: Since D is a subset of A, if w is a surjection, then D ¼ wða0 Þ for some a0 2 A. We must have either a0 2 D or a0 2 = D. If a0 2 D, then since D ¼ wða0 Þ, we must have a0 2 wða0 Þ, contrary to the definition of D. Similarly, if a0 2 = D, then a0 2 = wða0 Þ so that a0 2 D, which is also a contradiction. Therefore, w cannot be a surjection. Q.E.D. Cantor’s Theorem implies that there is an unending progression of larger and larger sets. In particular, it implies that the collection PðNÞ of all subsets of the natural numbers N is uncountable. C01 12/08/2010 12:3:27 22 Page 22 CHAPTER 1 PRELIMINARIES Exercises for Section 1.3 1. Prove that a nonempty set T1 is finite if and only if there is a bijection from T1 onto a finite set T2. 2. Prove parts (b) and (c) of Theorem 1.3.4. 3. Let S :¼ {1, 2} and T :¼ {a, b, c}. (a) Determine the number of different injections from S into T. (b) Determine the number of different surjections from T onto S. 4. Exhibit a bijection between N and the set of all odd integers greater than 13. 5. Give an explicit definition of the bijection f from N onto Z described in Example 1.3.7(b). 6. Exhibit a bijection between N and a proper subset of itself. 7. Prove that a set T1 is denumerable if and only if there is a bijection from T1 onto a denumerable set T2. 8. Give an example of a countable collection of finite sets whose union is not finite. 9. Prove in detail that if S and T are denumerable, then S [ T is denumerable. 10. (a) If (m, n) is the 6th point down the 9th diagonal of the array in Figure 1.3.1, calculate its number according to the counting method given for Theorem 1.3.8. (b) Given that h(m, 3) ¼ 19, find m. 11. Determine the number of elements in PðSÞ, the collection of all subsets of S, for each of the following sets: (a) S :¼ {1, 2}, (b) S :¼ {1, 2, 3}, (c) S :¼ {1, 2, 3, 4}. Be sure to include the empty set and the set S itself in PðSÞ. 12. Use Mathematical Induction to prove that if the set S has n elements, then PðSÞ has 2n elements. 13. Prove that the collection F ðNÞ of all finite subsets of N is countable. C02 12/09/2010 15:11:23 Page 23 CHAPTER 2 THE REAL NUMBERS In this chapter we will discuss the essential properties of the real number system R. Although it is possible to give a formal construction of this system on the basis of a more primitive set (such as the set N of natural numbers or the set Q of rational numbers), we have chosen not to do so. Instead, we exhibit a list of fundamental properties associated with the real numbers and show how further properties can be deduced from them. This kind of activity is much more useful in learning the tools of analysis than examining the logical difficulties of constructing a model for R. The real number system can be described as a ‘‘complete ordered field,’’ and we will discuss that description in considerable detail. In Section 2.1, we first introduce the ‘‘algebraic’’ properties—often called the ‘‘field’’ properties in abstract algebra—that are based on the two operations of addition and multiplication. We continue the section with the introduction of the ‘‘order’’ properties of R and we derive some consequences of these properties and illustrate their use in working with inequalities. The notion of absolute value, which is based on the order properties, is discussed in Section 2.2. In Section 2.3, we make the final step by adding the crucial ‘‘completeness’’ property to the algebraic and order properties of R. It is this property, which was not fully understood until the late nineteenth century, that underlies the theory of limits and continuity and essentially all that follows in this book. The rigorous development of real analysis would not be possible without this essential property. In Section 2.4, we apply the Completeness Property to derive several fundamental results concerning R, including the Archimedean Property, the existence of square roots, and the density of rational numbers in R. We establish, in Section 2.5, the Nested Interval Property and use it to prove the uncountability of R. We also discuss its relation to binary and decimal representations of real numbers. Part of the purpose of Sections 2.1 and 2.2 is to provide examples of proofs of elementary theorems from explicitly stated assumptions. Students can thus gain experience in writing formal proofs before encountering the more subtle and complicated arguments related to the Completeness Property and its consequences. However, students who have previously studied the axiomatic method and the technique of proofs (perhaps in a course on abstract algebra) can move to Section 2.3 after a cursory look at the earlier sections. A brief discussion of logic and types of proofs can be found in Appendix A at the back of the book. Terms such as ‘‘contrapositive’’ and ‘‘converse’’ are explained there and several proofs are examined in detail. Section 2.1 The Algebraic and Order Properties of R We begin with a brief discussion of the ‘‘algebraic structure’’ of the real number system. We will give a short list of basic properties of addition and multiplication from which all other algebraic properties can be derived as theorems. In the terminology of abstract 23 C02 12/09/2010 15:11:24 24 Page 24 CHAPTER 2 THE REAL NUMBERS algebra, the system of real numbers is a ‘‘field’’ with respect to addition and multiplication. The basic properties listed in 2.1.1 are known as the field axioms. A binary operation associates with each pair (a, b) a unique element B(a, b), but we will use the conventional notations of a þ b and a b when discussing the properties of addition and multiplication. 2.1.1 Algebraic Properties of R On the set R of real numbers there are two binary operations, denoted by þ and and called addition and multiplication, respectively. These operations satisfy the following properties: (A1) a þ b ¼ b þ a for all a, b in R (commutative property of addition); (A2) ða þ bÞ þ c ¼ a þ ðb þ cÞ for all a, b, c in R (associative property of addition); (A3) there exists an element 0 in R such that 0 þ a ¼ a and a þ 0 ¼ a for all a in R (existence of a zero element); (A4) for each a in R there exists an element a in R such that a þ ðaÞ ¼ 0 and ðaÞ þ a ¼ 0 (existence of negative elements); (M1) a b ¼ b a for all a, b in R (commutative property of multiplication); (M2) ða bÞ c ¼ a ðb cÞ for all a, b, c in R (associative property of multiplication); (M3) there exists an element 1 in R distinct from 0 such that 1 a ¼ a and a 1 ¼ a for all a in R (existence of a unit element); (M4) for each a 6¼ 0 in R there exists an element 1=a in R such that a ð1=aÞ ¼ 1 and ð1=aÞ a ¼ 1 (existence of reciprocals); (D) a ðb þ cÞ ¼ ða bÞ þ ða cÞ and ðb þ cÞ a ¼ ðb aÞ þ ðc aÞ for all a, b, c in R (distributive property of multiplication over addition). These properties should be familiar to the reader. The first four are concerned with addition, the next four with multiplication, and the last one connects the two operations. The point of the list is that all the familiar techniques of algebra can be derived from these nine properties, in much the same spirit that the theorems of Euclidean geometry can be deduced from the five basic axioms stated by Euclid in his Elements. Since this task more properly belongs to a course in abstract algebra, we will not carry it out here. However, to exhibit the spirit of the endeavor, we will sample a few results and their proofs. We first establish the basic fact that the elements 0 and 1, whose existence were asserted in (A3) and (M3), are in fact unique. We also show that multiplication by 0 always results in 0. 2.1.2 Theorem (a) If z and a are elements in R with z þ a ¼ a, then z ¼ 0. (b) If u and b 6¼ 0 are elements in R with u b ¼ b, then u ¼ 1. (c) If a 2 R, then a 0 ¼ 0. Proof. (a) Using (A3), (A4), (A2), the hypothesis z þ a ¼ a, and (A4), we get z ¼ z þ 0 ¼ z þ ða þ ðaÞÞ ¼ ðz þ aÞ þ ðaÞ ¼ a þ ðaÞ ¼ 0: (b) Using (M3), (M4), (M2), the assumed equality u b ¼ b, and (M4) again, we get u ¼ u 1 ¼ u ðb ð1=bÞÞ ¼ ðu bÞ ð1=bÞ ¼ b ð1=bÞ ¼ 1: (c) We have (why?) a þ a 0 ¼ a 1 þ a 0 ¼ a ð1 þ 0Þ ¼ a 1 ¼ a: Therefore, we conclude from (a) that a 0 ¼ 0. Q.E.D. C02 12/09/2010 15:11:24 Page 25 2.1 THE ALGEBRAIC AND ORDER PROPERTIES OF R 25 We next establish two important properties of multiplication: the uniqueness of reciprocals and the fact that a product of two numbers is zero only when one of the factors is zero. 2.1.3 Theorem (a) If a 6¼ 0 and b in R are such that a b ¼ 1, then b ¼ 1=a. (b) If a b ¼ 0, then either a ¼ 0 or b ¼ 0. Proof. (a) Using (M3), (M4), (M2), the hypothesis a b ¼ 1, and (M3), we have b ¼ 1 b ¼ ðð1=aÞ aÞ b ¼ ð1=aÞ ða bÞ ¼ ð1=aÞ 1 ¼ 1=a: (b) It suffices to assume a 6¼ 0 and prove that b ¼ 0. (Why?) We multiply a b by 1=a and apply (M2), (M4), and (M3) to get ð1=aÞ ða bÞ ¼ ðð1=aÞ aÞ b ¼ 1 b ¼ b: Since a b ¼ 0, by 2.1.2(c) this also equals ð1=aÞ ða bÞ ¼ ð1=aÞ 0 ¼ 0: Thus we have b ¼ 0. Q.E.D. These theorems represent a small sample of the algebraic properties of the real number system. Some additional consequences of the field properties are given in the exercises. The operation of subtraction is defined by a b :¼ a þ ðbÞ for a, b in R. Similarly, division is defined for a, b in R with b 6¼ 0 by a=b :¼ a ð1=bÞ. In the following, we will use this customary notation for subtraction and division, and we will use all the familiar properties of these operations. We will ordinarily drop the use of the dot to indicate multiplication and write ab for a b. Similarly, we will use the usual notation for exponents and write a2 for aa, a3 for ða2 Þa; and, in general, we define anþ1 :¼ ðan Þa for n 2 N. We agree to adopt the convention that a1 ¼ a. Further, if a 6¼ 0, we write a0 ¼ 1 and a1 for 1=a, and if n 2 N, we will write an for ð1=aÞn, when it is convenient to do so. In general, we will freely apply all the usual techniques of algebra without further elaboration. Rational and Irrational Numbers We regard the set N of natural numbers as a subset of R, by identifying the natural number n 2 N with the n-fold sum of the unit element 1 2 R. Similarly, we identify 0 2 Z with the zero element of 0 2 R, and we identify the n-fold sum of 1 with the integer n. Thus, we consider N and Z to be subsets of R. Elements of R that can be written in the form b=a where a; b 2 Z and a 6¼ 0 are called rational numbers. The set of all rational numbers in R will be denoted by the standard notation Q . The sum and product of two rational numbers is again a rational number (prove this), and moreover, the field properties listed at the beginning of this section can be shown to hold for Q. The fact that there are elements in R that are not in Q is not immediately apparent. In the sixth century B.C. the ancient Greek society of Pythagoreans discovered that the diagonal of a square with unit sides could not be expressed as a ratio of integers. In view of the Pythagorean Theorem for right triangles, this implies that the square of no rational number can equal 2. This discovery had a profound impact on the development of Greek mathematics. One consequence is that elements of R that are not in Q became known as irrational numbers, meaning that they are not ratios of integers. Although the word ‘‘irrational’’ in modern English usage has a quite different meaning, we shall adopt the standard mathematical usage of this term. C02 12/09/2010 15:11:25 26 Page 26 CHAPTER 2 THE REAL NUMBERS We will now prove that there does not exist a rational number whose square is 2. In the proof we use the notions of even and odd numbers. Recall that a natural number is even if it has the form 2n for some n 2 N, and it is odd if it has the form 2n 1 for some n 2 N. Every natural number is either even or odd, and no natural number is both even and odd. 2.1.4 Theorem There does not exist a rational number r such that r2 ¼ 2. Proof. Suppose, on the contrary, that p and q are integers such that ðp=qÞ2 ¼ 2. We may assume that p and q are positive and have no common integer factors other than 1. (Why?) Since p2 ¼ 2q2 , we see that p2 is even. This implies that p is also even (because if p ¼ 2n 1 is odd, then its square p2 ¼ 2ð2n2 2n þ 1Þ 1 is also odd). Therefore, since p and q do not have 2 as a common factor, then q must be an odd natural number. Since p is even, then p ¼ 2m for some m 2 N, and hence 4m2 ¼ 2q2 , so that 2m2 ¼ q2 . Therefore, q2 is even, and it follows that q is an even natural number. Since the hypothesis that ðp=qÞ2 ¼ 2 leads to the contradictory conclusion that q is both even and odd, it must be false. Q.E.D. The Order Properties of R The ‘‘order properties’’ of R refer to the notions of positivity and inequalities between real numbers. As with the algebraic structure of the system of real numbers, we proceed by isolating three basic properties from which all other order properties and calculations with inequalities can be deduced. The simplest way to do this is to identify a special subset of R by using the notion of ‘‘positivity.’’ 2.1.5 The Order Properties of R There is a nonempty subset P of R, called the set of positive real numbers, that satisfies the following properties: (i) (ii) (iii) If a, b belong to P, then a þ b belongs to P. If a, b belong to P, then ab belongs to P. If a belongs to R, then exactly one of the following holds: a 2 P; a ¼ 0; a 2 P: The first two conditions ensure the compatibility of order with the operations of addition and multiplication, respectively. Condition 2.1.5(iii) is usually called the Trichotomy Property, since it divides R into three distinct types of elements. It states that the set fa : a 2 Pg of negative real numbers has no elements in common with the set P of positive real numbers, and, moreover, the set R is the union of three disjoint sets. If a 2 P, we write a > 0 and say that a is a positive (or a strictly positive) real number. If a 2 P [ f0g, we write a 0 and say that a is a nonnegative real number. Similarly, if a 2 P, we write a < 0 and say that a is a negative (or a strictly negative) real number. If a 2 P [ f0g, we write a 0 and say that a is a nonpositive real number. The notion of inequality between two real numbers will now be defined in terms of the set P of positive elements. 2.1.6 Definition Let a, b be elements of R. (a) If a b 2 P, then we write a > b or b < a. (b) If a b 2 P [ f0g, then we write a b or b a. C02 12/09/2010 15:11:25 Page 27 2.1 THE ALGEBRAIC AND ORDER PROPERTIES OF R 27 The Trichotomy Property 2.1.5(iii) implies that for a; b 2 R exactly one of the following will hold: a > b; a ¼ b; a < b: Therefore, if both a b and b a, then a ¼ b. For notational convenience, we will write a<b<c to mean that both a < b and b < c are satisfied. The other ‘‘double’’ inequalities a b < c, a b c, and a < b c are defined in a similar manner. To illustrate how the basic Order Properties are used to derive the ‘‘rules of inequalities,’’ we will now establish several results that the reader has used in earlier mathematics courses. 2.1.7 Theorem Let a, b, c be any elements of R. (a) If a > b and b > c , then a > c. (b) If a > b, then a þ c > b þ c. (c) If a > b and c > 0, then ca > cb. If a > b and c < 0, then ca < cb. Proof. (a) If a b 2 P and b c 2 P, then 2.1.5(i) implies that ða bÞ þ ðb cÞ ¼ a c belongs to P. Hence a > c. (b) If a b 2 P, then ða þ cÞ ðb þ cÞ ¼ a b is in P. Thus a þ c > b þ c. (c) If a b 2 P and c 2 P, then ca cb ¼ cða bÞ is in P by 2.1.5(ii). Thus ca > cb when c > 0. On the other hand, if c < 0, then c 2 P, so that cb ca ¼ ðcÞða bÞ is in P. Thus cb > ca when c < 0. Q.E.D. It is natural to expect that the natural numbers are positive real numbers. This property is derived from the basic properties of order. The key observation is that the square of any nonzero real number is positive. 2.1.8 Theorem (a) If a 2 R and a 6¼ 0, then a2 > 0. (b) 1 > 0. (c) If n 2 N, then n > 0. Proof. (a) By the Trichotomy Property, if a 6¼ 0, then either a 2 P or a 2 P. If a 2 P, then by 2.1.5(ii), we have a2 ¼ a a 2 P. Also, if a 2 P, then a2 ¼ ðaÞðaÞ 2 P. We conclude that if a 6¼ 0, then a2 > 0. (b) Since 1 ¼ 12 , it follows from (a) that 1 > 0. (c) We use Mathematical Induction. The assertion for n ¼ 1 is true by (b). If we suppose the assertion is true for the natural number k, then k 2 P, and since 1 2 P, we have k þ 1 2 P by 2.1.5(i). Therefore, the assertion is true for all natural numbers. Q.E.D. It is worth noting that no smallest positive real number can exist. This follows by observing that if a > 0, then since 12 > 0 (why?), we have that 0 < 12 a < a: C02 12/09/2010 15:11:26 28 Page 28 CHAPTER 2 THE REAL NUMBERS Thus if it is claimed that a is the smallest positive real number, we can exhibit a smaller positive number 12 a. This observation leads to the next result, which will be used frequently as a method of proof. For instance, to prove that a number a 0 is actually equal to zero, we see that it suffices to show that a is smaller than an arbitrary positive number. 2.1.9 Theorem If a 2 R is such that 0 a < e for every e > 0, then a ¼ 0. Proof. Suppose to the contrary that a > 0. Then if we take e0 :¼ 12 a, we have 0 < e0 < a. Therefore, it is false that a < e for every e > 0 and we conclude that a ¼ 0. Q.E.D. Remark It is an exercise to show that if a 2 R is such that 0 a e for every e > 0, then a ¼ 0. The product of two positive numbers is positive. However, the positivity of a product of two numbers does not imply that each factor is positive. The correct conclusion is given in the next theorem. It is an important tool in working with inequalities. 2.1.10 Theorem If ab > 0, then either (i) (ii) a > 0 and b > 0, or a < 0 and b < 0. Proof. First we note that ab > 0 implies that a ¼ 6 0 and b ¼ 6 0. (Why?) From the Trichotomy Property, either a > 0 or a < 0. If a > 0, then 1=a > 0, and therefore b ¼ ð1=aÞ ðabÞ > 0. Similarly, if a < 0, then 1=a < 0, so that b ¼ ð1=aÞðabÞ < 0. Q.E.D. 2.1.11 Corollary If ab < 0, then either (i) (ii) a < 0 and b > 0, or a > 0 and b < 0. Inequalities We now show how the Order Properties presented in this section can be used to ‘‘solve’’ certain inequalities. The reader should justify each of the steps. 2.1.12 Examples (a) Determine the set A of all real numbers x such that 2x þ 3 6. We note that we havey x 2 A () 2x þ 3 6 () 2x 3 () x 32 : Therefore A ¼ x 2 R : x 32 . (b) Determine the set B :¼ fx 2 R : x2 þ x > 2g. We rewrite the inequality so that Theorem 2.1.10 can be applied. Note that x 2 B () x2 þ x 2 > 0 () ðx 1Þðx þ 2Þ > 0: Therefore, we either have (i) x 1 > 0 and x þ 2 > 0, or we have (ii) x 1 < 0 and x þ 2 < 0. In case (i) we must have both x > 1 and x > 2, which is satisfied if and only y The symbol () should be read ‘‘if and only if.’’ C02 12/09/2010 15:11:26 Page 29 2.1 THE ALGEBRAIC AND ORDER PROPERTIES OF R 29 if x > 1. In case (ii) we must have both x < 1 and x < 2, which is satisfied if and only if x < 2. We conclude that B ¼ fx 2 R : x > 1g [ fx 2 R : x < 2g. (c) Determine the set 2x þ 1 <1 : C :¼ x 2 R : xþ2 We note that x 2 C () 2x þ 1 x1 1 < 0 () < 0: xþ2 xþ2 Therefore we have either (i) x 1 < 0 and x þ 2 > 0, or (ii) x 1 > 0 and x þ 2 < 0. (Why?) In case (i) we must have both x < 1 and x > 2, which is satisfied if and only if 2 < x < 1. In case (ii), we must have both x > 1 and x < 2, which is never satisfied. & We conclude that C ¼ fx 2 R : 2 < x < 1g. The following examples illustrate the use of the Order Properties of R in establishing certain inequalities. The reader should verify the steps in the arguments by identifying the properties that are employed. It should be noted that the existence of square roots of positive numbers has not yet been established; however, we assume the existence of these roots for the purpose of these examples. (The existence of square roots will be discussed in Section 2.4.) 2.1.13 Examples (a) Let a 0 and b 0. Then ð1Þ a < b () a2 < b2 () pffiffiffi pffiffiffi a< b We consider the case where a > 0 and b > 0, leaving the case a ¼ 0 to the reader. It follows from 2.1.5(i) that a þ b > 0. Since b2 a2 ¼ ðb aÞðb þ aÞ, it follows from 2.1.7(c) that b a > 0 implies that b2 a2 > 0. Also, it follows from 2.1.10 that b2 a2 > 0 implies that b a > 0. pffiffiffi pffiffiffi2 pffiffiffi pffiffiffi 2 If a > 0 and b > 0, then a > 0 and b > 0. Since a ¼ ð aÞ and b ¼ bpffiffiffi, the second implication is a consequence of the first one when a and b are replaced by a and pffiffiffi b, respectively. We also leave it to the reader to show that if a 0 and b 0, then pffiffiffi pffiffiffi 0 ð1 Þ a b () a2 b2 () a b 1 (b) If a and b are positive pffiffiffiffiffi real numbers, then their arithmetic mean is 2 ða þ bÞ and their geometric mean is ab. The Arithmetic-Geometric Mean Inequality for a, b is pffiffiffiffiffi ð2Þ ab 12 ða þ bÞ with equality occurring if and only if a ¼ b. pffiffiffi pffiffiffi To prove this, note that if a > 0; b > 0, and a 6¼ b, then a > 0; b > 0, and pffiffiffi pffiffiffi2 pffiffiffi pffiffiffi a 6¼ b. (Why?) Therefore it follows from 2.1.8(a) that a b > 0. Expanding this square, we obtain pffiffiffiffiffi a 2 ab þ b > 0; whence it follows that pffiffiffiffiffi ab < 12 ða þ bÞ: C02 12/09/2010 15:11:27 30 Page 30 CHAPTER 2 THE REAL NUMBERS Therefore (2) holds (with strict inequality) when a 6¼ b. Moreover, if a ¼ bð> 0Þ, then both sides of (2) equal a, so (2) becomes an equality. This proves that (2) holds for a > 0; b > 0. pffiffiffiffiffi On the other hand, suppose that a > 0; b > 0 and that ab ¼ 12 ða þ bÞ. Then, squaring both sides and multiplying by 4, we obtain 4ab ¼ ða þ bÞ2 ¼ a2 þ 2ab þ b2 ; whence it follows that 0 ¼ a2 2ab þ b2 ¼ ða bÞ2 : But this equality implies that a ¼ b. (Why?) Thus, equality in (2) implies that a ¼ b. Remark The general Arithmetic-Geometric Mean Inequality for the positive real numbers a1 ; a2 ; . . . ; an is ða1 a2 an Þ1=n ð3Þ a1 þ a2 þ þ an n with equality occurring if and only if a1 ¼ a2 ¼ ¼ an . It is possible to prove this more general statement using Mathematical Induction, but the proof is somewhat intricate. A more elegant proof that uses properties of the exponential function is indicated in Exercise 8.3.9 in Chapter 8. (c) Bernoulli’s Inequality. If x > 1, then ð1 þ xÞn 1 þ nx ð4Þ for all n2N The proof uses Mathematical Induction. The case n ¼ 1 yields equality, so the assertion is valid in this case. Next, we assume the validity of the inequality (4) for k 2 N and will deduce it for k þ 1. Indeed, the assumptions that ð1 þ xÞk 1 þ kx and that 1 þ x > 0 imply (why?) that ð1 þ xÞkþ1 ¼ ð1 þ xÞk ð1 þ xÞ ð1 þ kxÞ ð1 þ xÞ ¼ 1 þ ðk þ 1Þx þ kx2 1 þ ðk þ 1Þx: Thus, inequality (4) holds for n ¼ k þ 1. Therefore, (4) holds for all n 2 N. & Exercises for Section 2.1 1. If a; b 2 R, prove the following. (a) If a þ b ¼ 0, then b ¼ a, (c) ð1Þa ¼ a, (b) (d) ðaÞ ¼ a, ð1Þð1Þ ¼ 1. 2. Prove that if a; b 2 R, then (a) ða þ bÞ ¼ ðaÞ þ ðbÞ, (c) 1=ðaÞ ¼ ð1=aÞ, (b) (d) ðaÞ ðbÞ ¼ a b, ða=bÞ ¼ ðaÞ=b if b 6¼ 0. 3. Solve the following equations, justifying each step by referring to an appropriate property or theorem. (a) 2x þ 5 ¼ 8; (b) x2 ¼ 2x; 2 (c) x 1 ¼ 3; (d) ðx 1Þðx þ 2Þ ¼ 0: C02 12/09/2010 15:11:28 Page 31 2.1 THE ALGEBRAIC AND ORDER PROPERTIES OF R 31 4. If a 2 R satisfies a a ¼ a, prove that either a ¼ 0 or a ¼ 1. 5. If a 6¼ 0 and b 6¼ 0, show that 1=ðabÞ ¼ ð1=aÞð1=bÞ. 6. Use the argument in the proof of Theorem 2.1.4 to show that there does not exist a rational number s such that s2 ¼ 6. 7. Modify the proof of Theorem 2.1.4 to show that there does not exist a rational number t such that t2 ¼ 3. 8. (a) Show that if x, y are rational numbers, then x þ y and xy are rational numbers. (b) Prove that if x is a rational number and y is an irrational number, then x þ y is an irrational number. If, in addition, x 6¼ 0, then show that xy is an irrational number. pffiffiffi 9. Let K :¼ s þ t 2 : s; t 2 Q . Show that K satisfies the following: (a) If x1 ; x2 2 K, then x1 þ x2 2 K and x1 x2 2 K. (b) If x 6¼ 0 and x 2 K, then 1=x 2 K. (Thus the set K is a subfield of R. With the order inherited from R, the set K is an ordered field that lies between Q and R.) 10. (a) If a < b and c d, prove that a þ c < b þ d. (b) If 0 < a < b and 0 c d, prove that 0 ac bd. 11. (a) Show that if a > 0, then 1=a > 0 and 1=ð1=aÞ ¼ a. (b) Show that if a < b, then a < 12 ða þ bÞ < b. 12. Let a, b, c, d be numbers satisfying 0 < a < b and c < d < 0. Give an example where ac < bd, and one where bd < ac. 13. If a; b 2 R, show that a2 þ b2 ¼ 0 if and only if a ¼ 0 and b ¼ 0. 14. If 0 a < b, show that a2 ab < b2 . Show by example that it does not follow that a2 < ab < b2 . pffiffiffiffiffi 15. If 0 < a < b, show that (a) a < ab < b, and (b) 1=b < 1=a. 16. Find all real numbers x that satisfy the following inequalities. (b) 1 < x2 < 4; (a) x2 > 3x þ 4; (c) 1=x < x; (d) 1=x < x2 : 17. Prove the following form of Theorem 2.1.9: If a 2 R is such that 0 a e for every e > 0, then a ¼ 0. 18. Let a; b 2 R, and suppose that for every e > 0 we have a b þ e. Show that a b. 19. Prove that a ¼ b. 1 2 ða þ bÞ 2 12 ða2 þ b2 Þ for all a; b 2 R. Show that equality holds if and only if 20. (a) If 0 < c < 1, show that 0 < c2 < c < 1. (b) If 1 < c, show that 1 < c < c2 . 21. (a) Prove there is no n 2 N such that 0 < n < 1. (Use the Well-Ordering Property of N.) (b) Prove that no natural number can be both even and odd. 22. (a) If c > 1, show that cn c for all n 2 N, and that cn > c for n > 1. (b) If 0 < c < 1, show that cn c for all n 2 N, and that cn < c for n > 1. 23. If a > 0; b > 0; and n 2 N, show that a < b if and only if an < bn . [Hint: Use Mathematical Induction.] 24. (a) If c > 1 and m; n 2 N, show that cm > cn if and only if m > n. (b) If 0 < c < 1 and m; n 2 N, show that cm < cn if and only if m > n. 25. Assuming the existence of roots, show that if c > 1, then c1=m < c1=n if and only if m > n. 26. Use Mathematical Induction to show that if a 2 R and m; n 2 N, then amþn ¼ am an and ðam Þ ¼ amn . C02 12/09/2010 15:11:29 32 Page 32 CHAPTER 2 THE REAL NUMBERS Section 2.2 Absolute Value and the Real Line From the Trichotomy Property 2.1.5(iii), we are assured that if a 2 R and a 6¼ 0, then exactly one of the numbers a and a is positive. The absolute value of a 6¼ 0 is defined to be the positive one of these two numbers. The absolute value of 0 is defined to be 0. 2.2.1 Definition The absolute value of a real number a, denoted by jaj, is defined by 8 < a if a > 0; jaj :¼ 0 if a ¼ 0; : a if a < 0: For example, j5j ¼ 5 and j 8j ¼ 8. We see from the definition that jaj 0 for all a 2 R, and that jaj ¼ 0 if and only if a ¼ 0. Also j aj ¼ jaj for all a 2 R. Some additional properties are as follows. 2.2.2 Theorem (a) jabj ¼ jajjbj for all a; b 2 R: (b) jaj2 ¼ a2 for all a 2 R. (c) If c 0, then jaj c if and only if c a c. (d) jaj a jaj for all a 2 R. Proof. (a) If either a or b is 0, then both sides are equal to 0. There are four other cases to consider. If a > 0; b > 0, then ab > 0, so that jabj ¼ ab ¼ jajjbj. If a > 0; b < 0, then ab < 0, so that jabj ¼ ab ¼ aðbÞ ¼ jajjbj. The remaining cases are treated similarly. (b) Since a2 0, we have a2 ¼ ja2 j ¼ jaaj ¼ jajjaj ¼ jaj2 . (c) If jaj c, then we have both a c and a c (why?), which is equivalent to c a c. Conversely, if c a c, then we have both a c and a c (why?), so that jaj c. (d) Take c ¼ jaj in part (c). Q.E.D. The following important inequality will be used frequently. 2.2.3 Triangle Inequality If a; b 2 R, then ja þ bj jaj þ jbj. Proof. From 2.2.2(d), we have jaj a jaj and jbj b jbj. On adding these inequalities, we obtain ðjaj þ jbjÞ a þ b jaj þ jbj: Hence, by 2.2.2(c) we have ja þ bj jaj þ jbj. Q.E.D. It can be shown that equality occurs in the Triangle Inequality if and only if ab > 0, which is equivalent to saying that a and b have the same sign. (See Exercise 2.) There are many useful variations of the Triangle Inequality. Here are two. 2.2.4 Corollary If a; b 2 R, then (a) jaj jbj ja bj; (b) ja bj jaj þ jbj: Proof. (a) We write a ¼ a b þ b and then apply the Triangle Inequality to get jaj ¼ jða bÞ þ bj ja bj þ jbj. Now subtract jbj to get jaj jbj ja bj. Similarly, C02 12/09/2010 15:11:29 Page 33 2.2 ABSOLUTE VALUE AND THE REAL LINE 33 from jbj ¼ jb a þ aj jb aj þ jaj, we obtain ja bj ¼ jb aj jaj jbj. If we combine these two inequalities, using 2.2.2(c), we get the inequality in (a). (b) Replace b in the Triangle Inequality by b to get ja bj jaj þ j bj. Since j bj ¼ jbj we obtain the inequality in (b). Q.E.D. A straightforward application of Mathematical Induction extends the Triangle Inequality to any finite number of elements of R. 2.2.5 Corollary If a1 ; a2 ; . . . ; an are any real numbers, then ja1 þ a2 þ þ an j ja1 j þ ja2 j þ þ jan j: The following examples illustrate how the properties of absolute value can be used. 2.2.6 Examples (a) Determine the set A of x 2 R such that j2x þ 3j < 7. From a modification of 2.2.2(c) for the case of strict inequality, we see that x 2 A if and only if 7 < 2x þ 3 < 7, which is satisfied if and only if 10 < 2x < 4. Dividing by 2, we conclude that A ¼ fx 2 R : 5 < x < 2g. (b) Determine the set B :¼ fx 2 R : jx 1j < jxjg. One method is to consider cases so that the absolute value symbols can be removed. Here we take the cases ðiÞ x 1; ðiiÞ 0 x < 1; ðiiiÞ x < 0: (Why did we choose these three cases?) In case (i) the inequality becomes x 1 < x, which is satisfied without further restriction. Therefore all x such that x 1 belong to the set B. In case (ii), the inequality becomes ðx 1Þ < x, which requires that x > 12. Thus, this case contributes all x such that 12 < x < 1 to the set B. In case (iii), the inequality becomes ðx 1Þ < x, which is equivalent to 1 < 0. Since this statement is false, no value of x from case (iii) satisfies the inequality. Forming the union of the three cases, we conclude that B ¼ x 2 R : x > 12 . There is a second method of determining the set B based on the fact that a < b if and only if a2 < b2 when both a 0 and b 0. (See 2.1.13(a).) Thus, the inequality jx 1j < jxj is equivalent to the inequality jx 1j2 < jxj2. Since jaj2 ¼ a2 for any a by 2.2.2(b), we 2 2 1 can expand the square to obtain x1 2x þ 1 < x , which simplifies to x > 2. Thus, we again find that B ¼ x 2 R : x > 2 . This method of squaring can sometimes be used to advantage, but often a case analysis cannot be avoided when dealing with absolute values. A graphical view of the inequality is obtained by sketching the graphs of y ¼ jxj and y ¼ jx 1j, and interpreting the inequality jx 1j < jxj to mean that the graph of y ¼ jx 1j lies underneath the graph of y ¼ jxj. See Figure 2.2.1. y y=|x| y = | x – 1| x 1 2 Figure 2.2.1 1 jx 1j < jxj C02 12/09/2010 15:11:30 34 Page 34 CHAPTER 2 THE REAL NUMBERS (c) Solve the inequality j2x 1j x þ 1. There are two cases to consider. If x 12, then j2x 1j ¼ 2x þ 1 and the inequality becomes 2x þ 1 x þ 1, which is x 0. Thus case one gives us 0 x 12. For case two we assume x 12, which gives us the inequality 2x 1 x þ 1, or x 2. Since x 12, we obtain 12 x 2. Combining the two cases, we get 0 x 2. See Figure 2.2.2. y y=x+1 (2, 3) y = |2x – 1| (0, 1) –1 Figure 2.2.2 1 2 x j2x 1j x þ 1 (d) Let the function f be defined by f ðxÞ :¼ ð2x2 þ 3x þ 1Þ=ð2x 1Þ for 2 x < 3. Find a constant M such that j f ðxÞj M for all x satisfying 2 x 3. We consider separately the numerator and denominator of j f ðxÞj ¼ j2x2 þ 3x þ 1j : j2x 1j From the Triangle Inequality, we obtain j2x2 þ 3x þ 1j 2jxj2 þ 3jxj þ 1 2 32 þ 3 3 þ 1 ¼ 28 since jxj 3 for the x under consideration. Also, j2x 1j 2jxj 1 2 2 1 ¼ 3 since jxj 2 for the x under consideration. Thus, 1=j2x 1j 1=3 for x 2. (Why?) Therefore, for 2 x 3 we have j f ðxÞj 28=3. Hence we can take M ¼ 28=3. (Note that we have found one such constant M; evidently any number H > 28=3 will also satisfy j f ðxÞj H. It is also possible that 28=3 is not the smallest possible choice & for M.) The Real Line A convenient and familiar geometric interpretation of the real number system is the real line. In this interpretation, the absolute value jaj of an element a in R is regarded as the distance from a to the origin 0. More generally, the distance between elements a and b in R is ja bj. (See Figure 2.2.3.) Figure 2.2.3 The distance between a ¼ 2 and b ¼ 3 C02 12/09/2010 15:11:30 Page 35 2.2 ABSOLUTE VALUE AND THE REAL LINE 35 Later we will need precise language to discuss the notion of one real number being ‘‘close to’’ another. If a is a given real number, then saying that a real number x is ‘‘close to’’ a should mean that the distance jx aj between them is ‘‘small.’’ A context in which this idea can be discussed is provided by the terminology of neighborhoods, which we now define. 2.2.7 Definition Let a 2 R and e > 0. Then the e-neighborhood of a is the set V e ðaÞ :¼ fx 2 R : jx aj < eg. For a 2 R, the statement that x belongs to V e ðaÞ is equivalent to either of the statements (see Figure 2.2.4) e < x a < e () a e < x < a þ e: Figure 2.2.4 An e-neighborhood of a 2.2.8 Theorem Let a 2 R. If x belongs to the neighborhood V e ðaÞ for every e > 0, then x ¼ a. Proof. If a particular x satisfies jx aj < e for every e > 0, then it follows from 2.1.9 that jx aj ¼ 0, and hence x ¼ a. Q.E.D. 2.2.9 Examples (a) Let U :¼ fx : 0 < x < 1g. If a 2 U, then let e be the smaller of the two numbers a and 1 a. Then it is an exercise to show that V e ðaÞ is contained in U. Thus each element of U has some e-neighborhood of it contained in U. (b) If I :¼ fx : 0 x 1g, then for any e > 0, the e-neighborhood V e ð0Þ of 0 contains points not in I, and so V e ð0Þ is not contained in I. For example, the number xe :¼ e=2 is in V e ð0Þ but not in I. (c) If jx aj < e and jy bj < e, then the Triangle Inequality implies that jðx þ yÞ ða þ bÞj ¼ jðx aÞ þ ðy bÞj jx aj þ jy bj < 2e: Thus if x, y belong to the e-neighborhoods of a, b, respectively, then x þ y belongs to the & 2e-neighborhood of a þ b (but not necessarily to the e-neighborhood of a þ b). Exercises for Section 2.2 1. If a; b 2 R pand ffiffiffiffiffi b 6¼ 0, show that: (a) jaj ¼ a2 ; (b) ja=bj ¼ jaj=jbj: 2. If a; b 2 R, show that ja þ bj ¼ jaj þ jbj if and only if ab 0. 3. If x; y; z 2 R and x z, show that x y z if and only if jx yj þ jy zj ¼ jx zj. Interpret this geometrically. C02 12/09/2010 15:11:31 36 Page 36 CHAPTER 2 THE REAL NUMBERS 4. Show that jx aj < e if and only if a e < x < a þ e. 5. If a < x < b and a < y < b, show that jx yj < b a. Interpret this geometrically. 6. Find all x 2 R that satisfy the following inequalities: (a) j4x 5j 13; (b) jx2 1j 3: 7. Find all x 2 R that satisfy the equation jx þ 1j þ jx 2j ¼ 7. 8. Find all values of x that satisfy the following equations: (a) x þ 1 ¼ j2x 1j, (b) 2x 1 ¼ jx 5j. 9. Find all values of x that satisfy the following inequalities. Sketch graphs. (a) jx 2j x þ 1, (b) 3jxj 2 x: 10. Find all x 2 R that satisfy the following inequalities. (a) jx 1j > jx þ 1j; (b) jxj þ jx þ 1j < 2: 11. Sketch the graph of the equation y ¼ jxj jx 1j. 12. Find all x 2 R that satisfy the inequality 4 < jx þ 2j þ jx 1j < 5. 13. Find all x 2 R that satisfy both j2x 3j < 5 and jx þ 1j > 2 simultaneously. 14. Determine and sketch the set of pairs ðx; yÞ in R R that satisfy: (a) jxj ¼ jyj; (b) jxj þ jyj ¼ 1; (c) jxyj ¼ 2, (d) jxj jyj ¼ 2: 15. Determine and sketch the set of pairs (x, y) in R R that satisfy: (a) jxj jyj; (b) jxj þ jyj 1; (c) jxyj 2; (d) jxj jyj 2: 16. Let e > 0 and d > 0, and a 2 R. Show that V e ðaÞ \ V d ðaÞ and V e ðaÞ [ V d ðaÞ are g-neighborhoods of a for appropriate values of g. 17. Show that if a; b 2 R, and a 6¼ b, then there exist e-neighborhoods U of a and V of b such that U \ V ¼ ;. 18. Show that if a; b 2 R then (a) maxfa; bg ¼ 12 ða þ b þ ja bjÞ and minfa; bg ¼ 12 ða þ b ja bjÞ: (b) minfa; b; cg ¼ minfminfa; bg; cg: 19. Show that if a; b; c 2 R, then the ‘‘middle number’’ is midfa; b; cg ¼ minfmaxfa; bg; maxfb; cg; maxfc; agg. Section 2.3 The Completeness Property of R Thus far, we have discussed the algebraic properties and the order properties of the real number system R. In this section we shall present one more property of R that is often called the ‘‘Completeness Property.’’ The system Q of rational numbers also has pffiffithe ffi algebraic and order properties described in the preceding sections, but we have seen that 2 cannot be represented pffiffiffi as a rational number; therefore 2 does not belong to Q . This observation shows the necessity of an additional property to characterize the real number system. This additional property, the Completeness (or the Supremum) Property, is an essential property of R, and we will say that R is a complete ordered field. It is this special property that permits us to define and develop the various limiting procedures that will be discussed in the chapters that follow. There are several different ways to describe the Completeness Property. We choose to give what is probably the most efficient approach by assuming that each nonempty bounded subset of R has a supremum. C02 12/09/2010 15:11:31 Page 37 2.3 THE COMPLETENESS PROPERTY OF R 37 Suprema and Infima We now introduce the notions of upper bound and lower bound for a set of real numbers. These ideas will be of utmost importance in later sections. 2.3.1 Definition Let S be a nonempty subset of R. (a) The set S is said to be bounded above if there exists a number u 2 R such that s u for all s 2 S. Each such number u is called an upper bound of S. (b) The set S is said to be bounded below if there exists a number w 2 R such that w s for all s 2 S. Each such number w is called a lower bound of S. (c) A set is said to be bounded if it is both bounded above and bounded below. A set is said to be unbounded if it is not bounded. For example, the set S :¼ fx 2 R : x < 2g is bounded above; the number 2 and any number larger than 2 is an upper bound of S. This set has no lower bounds, so that the set is not bounded below. Thus it is unbounded (even though it is bounded above). If a set has one upper bound, then it has infinitely many upper bounds, because if u is an upper bound of S, then the numbers u þ 1; u þ 2; . . . are also upper bounds of S. (A similar observation is valid for lower bounds.) In the set of upper bounds of S and the set of lower bounds of S, we single out their least and greatest elements, respectively, for special attention in the following definition. (See Figure 2.3.1.) Figure 2.3.1 inf S and sup S 2.3.2 Definition Let S be a nonempty subset of R. (a) If S is bounded above, then a number u is said to be a supremum (or a least upper bound) of S if it satisfies the conditions: (1) u is an upper bound of S, and (2) if v is any upper bound of S, then u v. (b) If S is bounded below, then a number w is said to be an infimum (or a greatest lower bound) of S if it satisfies the conditions: (10 ) w is a lower bound of S, and (20 ) if t is any lower bound of S, then t w. It is not difficult to see that there can be only one supremum of a given subset S of R. (Then we can refer to the supremum of a set instead of a supremum.) For, suppose that u1 and u2 are both suprema of S. If u1 < u2 , then the hypothesis that u2 is a supremum implies that u1 cannot be an upper bound of S. Similarly, we see that u2 < u1 is not possible. Therefore, we must have u1 ¼ u2 . A similar argument can be given to show that the infimum of a set is uniquely determined. If the supremum or the infimum of a set S exists, we will denote them by sup S and inf S: C02 12/09/2010 15:11:32 38 Page 38 CHAPTER 2 THE REAL NUMBERS We also observe that if u0 is an arbitrary upper bound of a nonempty set S, then sup S u0 . This is because sup S is the least of the upper bounds of S. First of all, it needs to be emphasized that in order for a nonempty set S in R to have a supremum, it must have an upper bound. Thus, not every subset of R has a supremum; similarly, not every subset of R has an infimum. Indeed, there are four possibilities for a nonempty subset S of R: it can (i) (ii) (iii) (iv) have have have have both a supremum and an infimum, a supremum but no infimum, an infimum but no supremum, neither a supremum nor an infimum. We also wish to stress that in order to show that u ¼ sup S for some nonempty subset S of R, we need to show that both (1) and (2) of Definition 2.3.2(a) hold. It will be instructive to reformulate these statements. The definition of u ¼ sup S asserts that u is an upper bound of S such that u v for any upper bound v of S. It is useful to have alternative ways of expressing the idea that u is the ‘‘least’’ of the upper bounds of S. One way is to observe that any number smaller than u is not an upper bound of S. That is, if z < u, then z is not an upper bound of S. But to say that z is not an upper bound of S means there exists an element sz in S such that z < sz . Similarly, if e > 0, then u e is smaller than u and thus fails to be an upper bound of S. The following statements about an upper bound u of a set S are equivalent: (1) (2) (3) (4) if if if if v is any upper bound of S, then u v, z < u, then z is not an upper bound of S, z < u, then there exists sz 2 S such that z < sz , e > 0, then there exists se 2 S such that u e < se . Therefore, we can state two alternate formulations for the supremum. 2.3.3 Lemma A number u is the supremum of a nonempty subset S of R if and only if u satisfies the conditions: (1) s u for all s 2 S, (2) if v < u, then there exists s0 2 S such that v < s0 . For future work with limits, it is useful to have this condition expressed in terms of e > 0. This is done in the next lemma. 2.3.4 Lemma An upper bound u of a nonempty set S in R is the supremum of S if and only if for every e > 0 there exists an se 2 S such that u e < se . Proof. If u is an upper bound of S that satisfies the stated condition and if v < u, then we put e :¼ u v. Then e > 0, so there exists se 2 S such that v ¼ u e < se . Therefore, v is not an upper bound of S, and we conclude that u ¼ sup S. Conversely, suppose that u ¼ sup S and let e > 0. Since u e < u, then u e is not an upper bound of S. Therefore, some element se of S must be greater than u e; that is, u e < se . (See Figure 2.3.2.) Q.E.D. It is important to realize that the supremum of a set may or may not be an element of the set. Sometimes it is and sometimes it is not, depending on the particular set. We consider a few examples. C02 12/09/2010 15:11:32 Page 39 2.3 THE COMPLETENESS PROPERTY OF R Figure 2.3.2 39 u ¼ sup S 2.3.5 Examples (a) If a nonempty set S1 has a finite number of elements, then it can be shown that S1 has a largest element u and a least element w. Then u ¼ sup S1 and w ¼ inf S1, and they are both members of S1. (This is clear if S1 has only one element, and it can be proved by induction on the number of elements in S1; see Exercises 12 and 13.) (b) The set S2 :¼ fx : 0 x 1g clearly has 1 for an upper bound. We prove that 1 is its supremum as follows. If v < 1, there exists an element s0 2 S2 such that v < s0 . (Name one such element s0 .) Therefore v is not an upper bound of S2 and, since v is an arbitrary number v < 1, we conclude that sup S2 ¼ 1. It is similarly shown that inf S2 ¼ 0. Note that both the supremum and the infimum of S2 are contained in S2. (c) The set S3 :¼ fx : 0 < x < 1g clearly has 1 for an upper bound. Using the same argument as given in (b), we see that sup S3 ¼ 1. In this case, the set S3 does not contain its supremum. Similarly, inf S3 ¼ 0 is not contained in S3. & The Completeness Property of R It is not possible to prove on the basis of the field and order properties of R that were discussed in Section 2.1 that every nonempty subset of R that is bounded above has a supremum in R. However, it is a deep and fundamental property of the real number system that this is indeed the case. We will make frequent and essential use of this property, especially in our discussion of limiting processes. The following statement concerning the existence of suprema is our final assumption about R. Thus, we say that R is a complete ordered field. 2.3.6 The Completeness Property of R Every nonempty set of real numbers that has an upper bound also has a supremum in R. This property is also called the Supremum Property of R. The analogous property for infima can be deduced from the Completeness Property as follows. Suppose that S is a nonempty subset of R that is bounded below. Then the nonempty set S :¼ fs : s 2 Sg is bounded above, and the Supremum Property implies that u :¼ sup S exists in R. The reader should verify in detail that –u is the infimum of S. Exercises for Section 2.3 1. Let S1 :¼ fx 2 R : x 0g. Show in detail that the set S1 has lower bounds, but no upper bounds. Show that inf S1 ¼ 0. 2. Let S2 :¼ fx 2 R : x > 0g. Does S2 have lower bounds? Does S2 have upper bounds? Does inf S2 exist? Does sup S2 exist? Prove your statements. 3. Let S3 ¼ f1=n : n 2 NÞ. Show that sup S3 ¼ 1 and inf S3 0. (It will follow from the Archimedean Property in Section 2.4 that inf S3 ¼ 0.) 4. Let S4 :¼ f1 ð1Þn =n : n 2 N g. Find inf S4 and sup S4. C02 12/09/2010 15:11:33 40 Page 40 CHAPTER 2 THE REAL NUMBERS 5. Find the infimum and supremum, if they exist, of each of the following sets. (a) A :¼ fx 2 R : 2x þ 5 > 0g; (b) B :¼ x 2 R : x þ 2 x2 ; (c) C :¼ fx 2 R : x < 1=xg; (d) D :¼ x 2 R : x2 2x 5 < 0 : 6. Let S be a nonempty subset of R that is bounded below. Prove that inf S ¼ supfs : s 2 Sg. 7. If a set S R contains one of its upper bounds, show that this upper bound is the supremum of S. 8. Let S R be nonempty. Show that u 2 R is an upper bound of S if and only if the conditions t 2 R and t > u imply that t 2 = S. 9. Let S R be nonempty. Show that if u ¼ sup S, then for every number n 2 N the number u 1=n is not an upper bound of S, but the number u þ 1=n is an upper bound of S. (The converse is also true; see Exercise 2.4.3.) 10. Show that if A and B are bounded subsets of R, then A [ B is a bounded set. Show that supðA [ BÞ ¼ supfsup A; sup Bg. 11. Let S be a bounded set in R and let S0 be a nonempty subset of S. Show that inf S inf S0 sup S0 sup S. 12. Let S R and suppose that supðS [ fugÞ ¼ supfs ; ug. s :¼ sup S belongs to S. If u 2 = S, show that 13. Show that a nonempty finite set S R contains its supremum. [Hint: Use Mathematical Induction and the preceding exercise.] 14. Let S be a set that is bounded below. Prove that a lower bound w of S is the infimum of S if and only if for any e > 0 there exists t 2 S such that t < w þ e. Section 2.4 Applications of the Supremum Property We will now discuss how to work with suprema and infima. We will also give some very important applications of these concepts to derive fundamental properties of R. We begin with examples that illustrate useful techniques in applying the ideas of supremum and infimum. 2.4.1 Examples (a) It is an important fact that taking suprema and infima of sets is compatible with the algebraic properties of R. As an example, we present here the compatibility of taking suprema and addition. Let S be a nonempty subset of R that is bounded above, and let a be any number in R. Define the set a þ S :¼ fa þ s : s 2 Sg. We will prove that supða þ SÞ ¼ a þ sup S: If we let u :¼ sup S, then x u for all x 2 S, so that a þ x a þ u. Therefore, a þ u is an upper bound for the set a þ S ; consequently, we have supða þ SÞ a þ u. Now if v is any upper bound of the set a þ S, then a þ x v for all x 2 S. Consequently x v a for all x 2 S, so that v a is an upper bound of S. Therefore, u ¼ sup S v a, which gives us a þ u v. Since v is any upper bound of a þ S, we can replace v by supða þ SÞ to get a þ u supða þ SÞ. Combining these inequalities, we conclude that supða þ SÞ ¼ a þ u ¼ a þ sup S: For similar relationships between the suprema and infima of sets and the operations of addition and multiplication, see the exercises. C02 12/09/2010 15:11:33 Page 41 2.4 APPLICATIONS OF THE SUPREMUM PROPERTY 41 (b) If the suprema or infima of two sets are involved, it is often necessary to establish results in two stages, working with one set at a time. Here is an example. Suppose that A and B are nonempty subsets of R that satisfy the property: ab for all a 2 A and all b 2 B: We will prove that sup A inf B: For, given b 2 B, we have a b for all a 2 A. This means that b is an upper bound of A, so that sup A b. Next, since the last inequality holds for all b 2 B, we see that the number & sup A is a lower bound for the set B. Therefore, we conclude that sup A inf B. Functions The idea of upper bound and lower bound is applied to functions by considering the range of a function. Given a function f : D ! R, we say that f is bounded above if the set f ðDÞ ¼ f f ðxÞ : x 2 Dg is bounded above in R; that is, there exists B 2 R such that f ðxÞ B for all x 2 D. Similarly, the function f is bounded below if the set f (D) is bounded below. We say that f is bounded if it is bounded above and below; this is equivalent to saying that there exists B 2 R such that j f ðxÞj B for all x 2 D. The following example illustrates how to work with suprema and infima of functions. 2.4.2 Examples Suppose that f and g are real-valued functions with common domain D R. We assume that f and g are bounded. (a) If f ðxÞ gðxÞ for all x 2 D, then sup f ðDÞ sup gðDÞ, which is sometimes written: sup f ðxÞ sup gðxÞ: x2D x2D We first note that f ðxÞ gðxÞ sup gðDÞ, which implies that the number sup g(D) is an upper bound for f (D). Therefore, sup f ðDÞ sup gðDÞ. (b) We note that the hypothesis f ðxÞ gðxÞ for all x 2 D in part (a) does not imply any relation between sup f(D) and inf g(D). For example, if f ðxÞ :¼ x2 and gðxÞ :¼ x with D ¼ fx : 0 x 1g, then f ðxÞ gðxÞ for all x 2 D. However, we see that sup f ðDÞ ¼ 1 and inf gðDÞ ¼ 0. Since sup gðDÞ ¼ 1, the conclusion of (a) holds. (c) If f ðxÞ gðyÞ for all x, y 2 D, then we may conclude that sup f ðDÞ infgðDÞ, which we may write as: sup f ðxÞ inf gðyÞ: x2D y2D (Note that the functions in (b) do not satisfy this hypothesis.) The proof proceeds in two stages as in Example 2.4.l(b). The reader should write out & the details of the argument. Further relationships between suprema and infima of functions are given in the exercises. The Archimedean Property Because of your familiarity with the set R and the customary picture of the real line, it may seem obvious that the set N of natural numbers is not bounded in R. How can we prove this C02 12/09/2010 15:11:34 42 Page 42 CHAPTER 2 THE REAL NUMBERS ‘‘obvious’’ fact? In fact, we cannot do so by using only the Algebraic and Order Properties given in Section 2.1. Indeed, we must use the Completeness Property of R as well as the Inductive Property of N (that is, if n 2 N, then n þ 1 2 N). The absence of upper bounds for N means that given any real number x there exists a natural number n (depending on x) such that x < n. 2.4.3 Archimedean Property If x 2 R, then there exists nx 2 N such that x nx . Proof. If the assertion is false, then n x for all n 2 N; therefore, x is an upper bound of N. Therefore, by the Completeness Property, the nonempty set N has a supremum u 2 R. Subtracting 1 from u gives a number u 1, which is smaller than the supremum u of N. Therefore u 1 is not an upper bound of N, so there exists m 2 N with u 1 < m. Adding 1 gives u < m þ 1, and since m þ 1 2 N, this inequality contradicts the fact that u is an upper bound of N. Q.E.D. 2.4.4 Corollary If S :¼ f1=n : n 2 Ng, then inf S ¼ 0. Proof. Since S 6¼ ; is bounded below by 0, it has an infimum and we let w :¼ inf S. It is clear that w 0. For any e > 0, the Archimedean Property implies that there exists n 2 N such that 1=e < n, which implies 1=n < e. Therefore we have 0 w 1=n < e: But since e > 0 is arbitrary, it follows from Theorem 2.1.9 that w ¼ 0. Q.E.D. 2.4.5 Corollary If t > 0, there exists nt 2 N such that 0 < 1=nt < t. Proof. Since inf f1=n : n 2 N g ¼ 0 and t > 0, then t is not a lower bound for the set f1=n : n 2 Ng. Thus there exists nt 2 N such that 0 < 1=nt < t. Q.E.D. 2.4.6 Corollary If y > 0, there exists ny 2 N such that ny 1 y ny . Proof. The Archimedean Property ensures that the subset Ey :¼ fm 2 N : y < mg of N is not empty. By the Well-Ordering Property 1.2.1, Ey has a least element, which we denote by ny. Then ny 1 does not belong to Ey, and hence we have ny 1 y < ny . Q.E.D. Collectively, the Corollaries 2.4.4–2.4.6 are sometimes referred to as the Archimedean Property of R. The Existence of pffiffiffi 2 The importance of the Supremum Property lies in the fact that it guarantees the existence of real numbers under certain hypotheses. We shall make use of it in this way many times. At the moment, we shall illustrate this use by proving the existence of a positive real number x such that x2 ¼ 2; that is, the positive square root of 2. It was shown earlier (see Theorem 2.1.4) that such an x cannot be a rational number; thus, we will be deriving the existence of at least one irrational number. C02 12/09/2010 15:11:34 Page 43 2.4 APPLICATIONS OF THE SUPREMUM PROPERTY 43 2.4.7 Theorem There exists a positive real number x such that x2 ¼ 2. Proof. Let S :¼ s 2 R : 0 s; s2 < 2 . Since 1 2 S, the set is not empty. Also, S is bounded above by 2, because if t > 2, then t2 > 4 so that t 2 = S. Therefore the Supremum Property implies that the set S has a supremum in R, and we let x :¼ sup S. Note that x > 1. We will prove that x2 ¼ 2 by ruling out the other two possibilities: x2 < 2 and x2 > 2. First assume that x2 < 2. We will show that this assumption contradicts the fact that x ¼ sup S by finding an n 2 N such that x þ 1=n 2 S, thus implying that x is not an upper bound for S. To see how to choose n, note that 1=n2 1=n so that xþ 1 n 2 ¼ x2 þ 2x 1 1 þ x2 þ ð2x þ 1Þ: n n2 n Hence if we can choose n so that 1 ð2x þ 1Þ < 2 x2 ; n then we get ðx þ 1=nÞ2 < x2 þ ð2 x2 Þ ¼ 2. By assumption we have 2 x2 > 0, so that ð2 x2 Þ=ð2x þ 1Þ > 0. Hence the Archimedean Property (Corollary 2.4.5) can be used to obtain n 2 N such that 1 2 x2 < : n 2x þ 1 These steps can be reversed to show that for this choice of n we have x þ 1=n 2 S, which contradicts the fact that x is an upper bound of S. Therefore we cannot have x2 < 2. Now assume that x2 > 2. We will show that it is then possible to find m 2 N such that x 1=m is also an upper bound of S, contradicting the fact that x ¼ sup S. To do this, note that x 1 m 2 ¼ x2 2x 1 2x þ 2 > x2 : m m m Hence if we can choose m so that 2x < x2 2; m then ðx 1=mÞ2 > x2 ðx2 2Þ ¼ 2. Now by assumption we have x2 2 > 0, so that ðx2 2Þ=2x > 0. Hence, by the Archimedean Property, there exists m 2 N such that 1 x2 2 < : m 2x These steps can be reversed to show that for this choice of m we have ðx 1=mÞ2 > 2. Now if s 2 S, then s2 < 2 < ðx 1=mÞ2 , whence it follows from 2.1.13(a) that s < x 1=m. This implies that x1/m is an upper bound for S, which contradicts the fact that x ¼ sup S. Therefore we cannot have x2 > 2. Since the possibilities x2 < 2 and x2 > 2 have been excluded, we must have 2 x ¼ 2. Q.E.D. By slightly modifying the preceding argument, the reader can show that if a > 0, then there is a unique such that b2 ¼ a. We call b the positive square root of a and denote pffiffiffi b > 01=2 it by b ¼ a or b ¼ a . A slightly more complicated argument involving the binomial C02 12/09/2010 15:11:35 44 Page 44 CHAPTER 2 THE REAL NUMBERS theorem canpbe ffiffiffi formulated to establish the existence of a unique positive nth root of a, denoted by n a or a1=n, for each n 2 N. Remark If in the proof of Theorem 2.4.7 we replace the set S by the set of rational numbers T :¼ fr 2 Q : 0 r; r2 < 2g, the argument then gives the conclusion that y :¼ sup T satisfies y2 ¼ 2. Since we have seen in Theorem 2.1.4 that y cannot be a rational number, it follows that the set T that consists of rational numbers does not have a supremum belonging to the set Q . Thus the ordered field Q of rational numbers does not possess the Completeness Property. Density of Rational Numbers in R pffiffiffi We now know that there exists at least one irrational real number, namely 2. Actually there are ‘‘more’’ irrational numbers than rational numbers in the sense that the set of rational numbers is countable (as shown in Section 1.3), while the set of irrational numbers is uncountable (see Section 2.5). However, we next show that in spite of this apparent disparity, the set of rational numbers is ‘‘dense’’ in R in the sense that given any two real numbers there is a rational number between them (in fact, there are infinitely many such rational numbers). 2.4.8 The Density Theorem If x and y are any real numbers with x < y, then there exists a rational number r 2 Q such that x < r < y. Proof. It is no loss of generality (why?) to assume that x > 0. Since y x > 0, it follows from Corollary 2.4.5 that there exists n 2 N such that 1=n < y x. Therefore, we have nx þ 1 < ny. If we apply Corollary 2.4.6 to nx > 0, we obtain m 2 N with m 1 nx < m. Therefore, m nx þ 1 < ny, whence nx < m < ny. Thus, the rational number r :¼ m=n satisfies x < r < y. Q.E.D. To round out the discussion of the interlacing of rational and irrational numbers, we have the same ‘‘betweenness property’’ for the set of irrational numbers. 2.4.9 Corollary If x and y are real numbers with x < y, then there exists an irrational number z such that x < z < y. pffiffiffi pffiffiffi Proof. If we apply the Density Theorem 2.4.8 to the real numbers x= 2 and y= 2, we obtain a rational number r 6¼ 0 (why?) such that x y pffiffiffi < r < pffiffiffi : 2 2 pffiffiffi Then z :¼ r 2 is irrational (why?) and satisfies x < z < y. Q.E.D. Exercises for Section 2.4 1. Show that supf1 1=n : n 2 Ng ¼ 1. 2. If S :¼ f1=n 1=m : n; m 2 Ng, find inf S and sup S. 3. Let S R be nonempty. Prove that if a number u in R has the properties: (i) for every n 2 N the number u 1=n is not an upper bound of S, and (ii) for every number n 2 N the number u þ 1=n is an upper bound of S, then u ¼ sup S. (This is the converse of Exercise 2.3.9.) C02 12/09/2010 15:11:36 Page 45 2.4 APPLICATIONS OF THE SUPREMUM PROPERTY 45 4. Let S be a nonempty bounded set in R. (a) Let a > 0, and let aS :¼ fas : s 2 Sg. Prove that infðaSÞ ¼ a inf S; supðaSÞ ¼ a sup S: (b) Let b < 0 and let bS ¼ fbs : s 2 Sg. Prove that infðbSÞ ¼ b sup S; supðbSÞ ¼ b inf S: 5. Let S be a set of nonnegative real numbers that is bounded above and let T :¼ fx2 : x 2 Sg. Prove that if u ¼ sup S, then u2 ¼ sup T. Give an example that shows the conclusion may be false if the restriction against negative numbers is removed. 6. Let X be a nonempty set and let f : X ! R have bounded range in R. If a 2 R, show that Example 2.4.l(a) implies that supfa þ f ðxÞ : x 2 Xg ¼ a þ supf f ðxÞ : x 2 Xg: Show that we also have inffa þ f ðxÞ : x 2 Xg ¼ a þ inff f ðxÞ : x 2 Xg: 7. Let A and B be bounded nonempty subsets of R, and let A þ B :¼ fa þ b : a 2 A; b 2 Bg. Prove that supðA þ BÞ ¼ sup A þ sup B and infðA þ BÞ ¼ inf A þ inf B. 8. Let X be a nonempty set, and let f and g be defined on X and have bounded ranges in R. Show that supf f ðxÞ þ gðxÞ : x 2 Xg supf f ðxÞ : x 2 Xg þ supfgðxÞ : x 2 Xg and that inff f ðxÞ : x 2 Xg þ inffgðxÞ : x 2 Xg inff f ðxÞ þ gðxÞ : x 2 Xg: Give examples to show that each of these inequalities can be either equalities or strict inequalities. 9. Let X ¼ Y :¼ fx 2 R : 0 < x < 1g. Define h : X Y ! R by hðx; yÞ :¼ 2x þ y. (a) For each x 2 X, find f ðxÞ :¼ supfhðx; yÞ : y 2 Yg; then find inff f ðxÞ : x 2 Xg. (b) For each y 2 Y, find gðyÞ :¼ inffhðx; yÞ : x 2 Xg; then find supfgðyÞ : y 2 Yg. Compare with the result found in part (a). 10. Perform the computations in (a) and (b) of the preceding exercise for the function h : X Y ! R defined by 0 if x < y; hðx; yÞ :¼ 1 if x y: 11. Let X and Y be nonempty sets and let h : X Y ! R have bounded range in R. Let f : X ! R and g : Y ! R be defined by f ðxÞ :¼ supfhðx; yÞ : y 2 Yg; gðyÞ :¼ inffhðx; yÞ : x 2 Xg: Prove that supfgðyÞ : y 2 Yg inff f ðxÞ : x 2 Xg: We sometimes express this by writing sup inf hðx; yÞ inf sup hðx; yÞ: y x x y Note that Exercises 9 and 10 show that the inequality may be either an equality or a strict inequality. C02 12/09/2010 15:11:37 46 Page 46 CHAPTER 2 THE REAL NUMBERS 12. Let X and Y be nonempty sets and let h : X Y ! R have bounded range in R. Let F : X ! R and G : Y ! R be defined by FðxÞ :¼ supfhðx; yÞ : y 2 Yg; GðyÞ :¼ supfhðx; yÞ : x 2 Xg: Establish the Principle of the Iterated Suprema: supfhðx; yÞ : x 2 X; y 2 Yg ¼ supfFðxÞ : x 2 Xg ¼ supfGðyÞ : y 2 Yg We sometimes express this in symbols by sup hðx; yÞ ¼ sup sup hðx; yÞ ¼ sup sup hðx; yÞ: x;y x y y x 13. Given any x 2 R, show that there exists a unique n 2 Z such that n 1 x < n. 14. If y > 0, show that there exists n 2 N such that 1=2n < y. 15. Modify the argument in Theorem 2.4.7 to show that there exists a positive real number y such that y2 ¼ 3. 16. Modify the argument in Theorem 2.4.7 to show that if a > 0, then there exists a positive real number z such that z2 ¼ a. 17. Modify the argument in Theorem 2.4.7 to show that there exists a positive real number u such that u3 ¼ 2. 18. Complete the proof of the Density Theorem 2.4.8 by removing the assumption that x > 0. 19. If u > 0 is any real number and x < y, show that there exists a rational number r such that x < ru < y. (Hence the set fru : r 2 Q g is dense in R.) Section 2.5 Intervals The Order Relation on R determines a natural collection of subsets called ‘‘intervals.’’ The notations and terminology for these special sets will be familiar from earlier courses. If a; b 2 R satisfy a < b, then the open interval determined by a and b is the set ða; bÞ :¼ fx 2 R : a < x < bg: The points a and b are called the endpoints of the interval; however, the endpoints are not included in an open interval. If both endpoints are adjoined to this open interval, then we obtain the closed interval determined by a and b; namely, the set ½a; b :¼ fx 2 R : a x bg: The two half-open (or half-closed) intervals determined by a and b are [a, b), which includes the endpoint a, and (a, b], which includes the endpoint b. Each of these four intervals is bounded and has length defined by b a. If a ¼ b, the corresponding open interval is the empty set ða; aÞ ¼ ;, whereas the corresponding closed interval is the singleton set ½a; a ¼ fag. There are five types of unbounded intervals for which the symbols 1ðor þ 1Þ and 1 are used as notational convenience in place of the endpoints. The infinite open intervals are the sets of the form ða; 1Þ :¼ fx 2 R : x > ag and ð1; bÞ :¼ fx 2 R : x < bg: C02 12/09/2010 15:11:37 Page 47 2.5 INTERVALS 47 The first set has no upper bounds and the second one has no lower bounds. Adjoining endpoints gives us the infinite closed intervals: ½a; 1Þ :¼ fx 2 R : a xg ð1; b :¼ fx 2 R : x bg: and It is often convenient to think of the entire set R as an infinite interval; in this case, we write ð1; 1Þ :¼ R. No point is an endpoint of ð1; 1Þ. Warning It must be emphasized that 1 and 1 are not elements of R, but only convenient symbols. Characterization of Intervals An obvious property of intervals is that if two points x, y with x < y belong to an interval I, then any point lying between them also belongs to I. That is, if x < t < y, then the point t belongs to the same interval as x and y. In other words, if x and y belong to an interval I, then the interval [x, y] is contained in I. We now show that a subset of R possessing this property must be an interval. 2.5.1 Characterization Theorem If S is a subset of R that contains at least two points and has the property ð1Þ if x; y 2 S and x < y; then ½x; y S; then S is an interval. Proof. There are four cases to consider: (i) S is bounded, (ii) S is bounded above but not below, (iii) S is bounded below but not above, and (iv) S is neither bounded above nor below. Case (i): Let a :¼ inf S and b :¼ sup S. Then S ½a; b and we will show that ða; bÞ S. If a < z < b, then z is not a lower bound of S, so there exists x 2 S with x < z. Also, z is not an upper bound of S, so there exists y 2 S with z < y. Therefore z 2 ½x; y , so property (1) implies that z 2 S. Since z is an arbitrary element of ða; bÞ, we conclude that ða; bÞ S. Now if a 2 S and b 2 S, then S ¼ ½a; b . (Why?) If a 2 = S and b 2 = S, then S ¼ ða; bÞ. The other possibilities lead to either S ¼ ða; b or S ¼ ½a; bÞ. Case (ii): Let b :¼ sup S. Then S ð1; b and we will show that ð1; bÞ S. For, if z < b, then there exist x; y 2 S such that z 2 ½x; y S. (Why?) Therefore ð1; bÞ S. If b 2 S, then S ¼ ð1; b , and if b 2 = S, then S ¼ ð1; bÞ. Cases (iii) and (iv) are left as exercises. Q.E.D. Nested Intervals We say that a sequence of intervals I n ; n 2 N, is nested if the following chain of inclusions holds (see Figure 2.5.1): I1 I2 In I nþ1 For example, if I n :¼ ½0; 1=n for n 2 N, then I n I nþ1 for each n 2 N so that this sequence of intervals is nested. In this case, the element 0 belongs to all I n and the Archimedean Property 2.4.3 can be usedTto show that 0 is the only such common point. (Prove this.) We denote this by writing 1 n¼1 I n ¼ f0g. It is important to realize that, in general, a nested sequence of intervals need not have a common point. For example, if J n :¼ ð0; 1=nÞ for n 2 N, then this sequence of intervals is C02 12/09/2010 15:11:38 48 Page 48 CHAPTER 2 THE REAL NUMBERS Figure 2.5.1 Nested intervals nested, but there is no common point, since for every given x > 0, there exists (why?) m 2 N such that 1=m < x so that x 2 = J m . Similarly, the sequence of intervals K n :¼ ðn; 1Þ; n 2 N, is nested but has no common point. (Why?) However, it is an important property of R that every nested sequence of closed, bounded intervals does have a common point, as we will now prove. Notice that the completeness of R plays an essential role in establishing this property. 2.5.2 Nested Intervals Property If I n ¼ ½an ; bn ; n 2 N, is a nested sequence of closed bounded intervals, then there exists a number j 2 R such that j 2 I n for all n 2 N. Proof. Since the intervals are nested, we have I n I 1 for all n 2 N, so that an b1 for all n 2 N. Hence, the nonempty set fan : n 2 Ng is bounded above, and we let j be its supremum. Clearly an j for all n 2 N. We claim also that j bn for all n. This is established by showing that for any particular n, the number bn is an upper bound for the set fak : k 2 Ng. We consider two cases. (i) If n k, then since I n I k , we have ak bk bn . (ii) If k < n, then since I k I n , we have ak an bn . (See Figure 2.5.2.) Thus, we conclude that ak bn for all k, so that bn is an upper bound of the set fak : k 2 Ng. Hence, j bn for each n 2 N. Since an j bn for all n, we have j 2 I n for all n 2 N. Q.E.D. Figure 2.5.2 If k < n, then I n I k 2.5.3 Theorem If I n :¼ ½an ; bn ; n 2 N, is a nested sequence of closed, bounded intervals such that the lengths bn an of I n satisfy inf fbn an : n 2 N g ¼ 0; then the number j contained in In for all n 2 N is unique. Proof. If h :¼ inf fbn : n 2 N g, then an argument similar to the proof of 2.5.2 can be used to show that an h for all n, and hence that j h. In fact, it is an exercise (see Exercise 10) to show that x 2 I n for all n 2 N if and only if j x h. If we have inf fbn an : n 2 N g ¼ 0, C02 12/09/2010 15:11:38 Page 49 2.5 INTERVALS 49 then for any e > 0, there exists an m 2 N such that 0 h j bm am < e. Since this holds for all e > 0, it follows from Theorem 2.1.9 that h j ¼ 0. Therefore, we conclude that j ¼ h is the only point that belongs to In for every n 2 N. Q.E.D. The Uncountability of R The concept of a countable set was discussed in Section 1.3 and the countability of the set Q of rational numbers was established there. We will now use the Nested Interval Property to prove that the set R is an uncountable set. The proof was given by Georg Cantor in 1874 in the first of his papers on infinite sets. He later published a proof that used decimal representations of real numbers, and that proof will be given later in this section. 2.5.4 Theorem The set R of real numbers is not countable. Proof. We will prove that the unit interval I :¼ ½0; 1 is an uncountable set. This implies that the set R is an uncountable set, for if R were countable, then the subset I would also be countable. (See Theorem 1.3.9(a).) The proof is by contradiction. If we assume that I is countable, then we can enumerate the set as I ¼ fx1 ; x2 ; . . . ; xn ; . . .g. We first select a closed subinterval I 1 of I such that x1 2 = I 1 , then select a closed subinterval I2 of I1 such that x2 2 = I 2 , and so on. In this way, we obtain nonempty closed intervals I1 I2 In such that I n I and xn 2 = I n for all n. The Nested Intervals Property 2.5.2 implies that there exists a point j 2 I such that j 2 I n for all n. Therefore j 6¼ xn for all n 2 N, so the enumeration of I is not a complete listing of the elements of I, as claimed. Hence, I is an uncountable set. Q.E.D. The fact that the set R of real numbers is uncountable can be combined with the fact that the set Q of rational numbers is countable to conclude that the set RnQ of irrational numbers is uncountable. Indeed, since the union of two countable sets is countable (see 1.3.7(c)), if RnQ is countable, then since R ¼ Q [ ðRnQ Þ, we conclude that R is also a countable set, which is a contradiction. Therefore, the set of irrational numbers RnQ is an uncountable set. Note: The set of real numbers can also be divided into two subsets of numbers called algebraic numbers and transcendental numbers. A real number is called algebraic if it is a solution of a polynomial equation PðxÞ ¼ 0 where all the coefficients of the polynomial P are integers. A real number is called transcendental if it is not an algebraic number. It can be proved that the set of algebraic numbers is countably infinite, and consequently the set of transcendental numbers is uncountable. The numbers p and e are transcendental numbers, but the proofs of these facts are very deep. For an introduction to these topics, we refer the interested reader to the book by Ivan Niven listed in the References. y Binary Representations We will digress briefly to discuss informally the binary (and decimal) representations of real numbers. It will suffice to consider real numbers between 0 and 1, since the representations for other real numbers can then be obtained by adding a positive or negative number. y The remainder of this section can be omitted on a first reading. C02 12/09/2010 15:11:39 50 Page 50 CHAPTER 2 THE REAL NUMBERS If x 2 ½0; 1 , we will use a repeated bisection procedure to associate a sequence (an) of 0s and 1s as follows. If x 6¼ 12 belongs to the left subinterval 0; 12 we take a1 :¼ 0, while if x belongs to the right subinterval 12 ; 1 we take a1 ¼ 1. If x ¼ 12, then we may take a1 to be either 0 or 1. In any case, we have a1 a1 þ 1 : x 2 2 We now bisect the interval 12 a1 ; 12 ða1 þ 1Þ . If x is not the bisection point and belongs to the left subinterval we take a2 :¼ 0, and if x belongs to the right subinterval we take a2 :¼ 1. If x ¼ 14 or x ¼ 34, we can take a2 to be either 0 or 1. In any case, we have a1 a2 a1 a2 þ 1 þ 2x þ : 2 2 2 22 We continue this bisection procedure, assigning at the nth stage the value an :¼ 0 if x is not the bisection point and lies in the left subinterval, and assigning the value an :¼ 1 if x lies in the right subinterval. In this way we obtain a sequence (an) of 0s or 1s that correspond to a nested sequence of intervals containing the point x. For each n, we have the inequality ð2Þ a1 a2 an a1 a2 an þ 1 þ þþ n x þ 2 þþ : 2n 2 22 2 2 2 If x is the bisection point at the nth stage, then x ¼ m=2n with m odd. In this case, we may choose either the left or the right subinterval; however, once this subinterval is chosen, then all subsequent subintervals in the bisection procedure are determined. [For instance, if we choose the left subinterval so that an ¼ 0, then x is the right endpoint of all subsequent subintervals, and hence ak ¼ 1 for all k n þ 1. On the other hand, if we choose the right subinterval so that an ¼ 1, then x is the left endpoint of all subsequent subintervals, and hence ak ¼ 0 for all k n þ 1. For example, if x ¼ 34, then the two possible sequences for x are 1, 0, 1, 1, 1, . . . and 1, 1, 0, 0, 0, . . . .] To summarize: If x 2 ½0; 1 , then there exists a sequence (an) of 0s and 1s such that inequality (2) holds for all n 2 N. In this case we write ð3Þ x ¼ ð:a1 a2 an Þ2 ; and call (3) a binary representation of x. This representation is unique except when x ¼ m=2n for m odd, in which case x has the two representations x ¼ ð:a1 a2 an1 1000 Þ2 ¼ ð:a1 a2 an1 0111 Þ2 ; one ending in 0s and the other ending in 1s. Conversely, each sequence of 0s and 1s is the binary representation of a unique real number in [0,1]. The inequality corresponding to (2) determines a closed interval with length 1=2n and the sequence of these intervals is nested. Therefore, Theorem 2.5.3 implies that there exists a unique real number x satisfying (2) for every n 2 N. Consequently, x has the binary representation ð:a1 a2 an Þ2 . Remark The concept of binary representation is extremely important in this era of digital computers. A number is entered in a digital computer on ‘‘bits,’’ and each bit can be put in one of two states—either it will pass current or it will not. These two states correspond to the values 1 and 0, respectively. Thus, the binary representation of a number can be stored in a digital computer on a string of bits. Of course, in actual practice, since only finitely many bits can be stored, the binary representations must be truncated. If n binary digits are C02 12/09/2010 15:11:39 Page 51 2.5 INTERVALS 51 used for a number x 2 ½0; 1 , then the accuracy is at most 1=2n. For example, to assure fourdecimal accuracy, it is necessary to use at least 15 binary digits (or 15 bits). Decimal Representations Decimal representations of real numbers are similar to binary representations, except that we subdivide intervals into ten equal subintervals instead of two. Thus, given x 2 ½0; 1 , if we subdivide [0,1] into ten equal subintervals, then x belongs to a subinterval ½b1 =10; ðb1 þ 1Þ=10 for some integer b1 in f0; 1; . . . ; 9g. Proceeding as in the binary case, we obtain a sequence (bn) of integers with 0 bn 9 for all n 2 N such that x satisfies ð4Þ b1 b2 bn b1 b2 bn þ 1 þ þ þ þ n x þ þ : 10n 10 102 10 10 102 In this case we say that x has a decimal representation given by x ¼ :b1 b2 bn : If x 1 and if B 2 N is such that B x < B þ 1, then x ¼ B:b1 b2 bn where the decimal representation of x B 2 ½0; 1 is as above. Negative numbers are treated similarly. The fact that each decimal determines a unique real number follows from Theorem 2.5.3, since each decimal specifies a nested sequence of intervals with lengths 1=10n. The decimal representation of x 2 ½0; 1 is unique except when x is a subdivision point at some stage, which can be seen to occur when x ¼ m=10n for some m; n 2 N; 1 m 10n . (We may also assume that m is not divisible by 10.) When x is a subdivision point at the nth stage, one choice for bn corresponds to selecting the left subinterval, which causes all subsequent digits to be 9, and the other choice corresponds to selecting the right subinterval, which causes all subsequent digits to be 0. [For example, if x ¼ 12 then x ¼ :4999 ¼ :5000 , and if y ¼ 38=100 then y ¼ :37999 ¼ :38000 .] Periodic Decimals A decimal B:b1 b2 bn is said to be periodic (or to be repeating), if there exist k; n 2 N such that bn ¼ bnþm for all n k. In this case, the block of digits bk bkþ1 bkþm1 is repeated once the kth digit is reached. The smallest number m with this property is called the period of the decimal. For example, 19=88 ¼ :2159090 90 has period m ¼ 2 with repeating block 90 starting at k ¼ 4. A terminating decimal is a periodic decimal where the repeated block is simply the digit 0. We will give an informal proof of the assertion: A positive real number is rational if and only if its decimal representation is periodic. Suppose that x ¼ p=q where p; q 2 N have no common integer factors. For convenience we will also suppose that 0 < p < q. We note that the process of ‘‘long division’’ of q into p gives the decimal representation of p=q. Each step in the division process produces a remainder that is an integer from 0 to q 1. Therefore, after at most q steps, some remainder will occur a second time and, at that point, the digits in the quotient will begin to repeat themselves in cycles. Hence, the decimal representation of such a rational number is periodic. Conversely, if a decimal is periodic, then it represents a rational number. The idea of the proof is best illustrated by an example. Suppose that x ¼ 7:31414 14 . We multiply by a power of 10 to move the decimal point to the first repeating block; here C02 12/09/2010 15:11:40 52 Page 52 CHAPTER 2 THE REAL NUMBERS obtaining 10x ¼ 73:1414 . We now multiply by a power of 10 to move one block to the left of the decimal point; here getting 1000x ¼ 7314:1414 . We now subtract to obtain an integer; here getting 1000x 10x ¼ 7314 73 ¼ 7241, whence x ¼ 7241=990, a rational number. Cantor’s Second Proof We will now give Cantor’s second proof of the uncountability of R. This is the elegant ‘‘diagonal’’ argument based on decimal representations of real numbers. 2.5.5 Theorem The unit interval ½0; 1 :¼ fx 2 R : 0 x 1g is not countable. Proof. The proof is by contradiction. We will use the fact that every real number x 2 ½0; 1 has a decimal representation x ¼ 0:b1 b2 b3 , where bi ¼ 0; 1; . . . ; 9. Suppose that there is an enumeration x1 ; x2 ; x3 of all numbers in [0,1], which we display as: x1 ¼ 0:b11 b12 b13 b1n ; x2 ¼ 0:b21 b22 b23 b2n ; x3 ¼ 0:b31 b32 b33 b3n ; xn ¼ 0:bn1 bn2 bn3 bnn ; We now define a real number y :¼ 0:y1 y2 y3 yn by setting y1 :¼ 2 if b11 5 and y1 :¼ 7 if b11 4; in general, we let 2 if bnn 5; yn :¼ 7 if bnn 4: Then y 2 ½0; 1 . Note that the number y is not equal to any of the numbers with two decimal representations, since yn 6¼ 0; 9 for all n 2 N. Further, since y and xn differ in the nth decimal place, then y 6¼ xn for any n 2 N. Therefore, y is not included in the enumeration of [0,1], contradicting the hypothesis. Q.E.D. Exercises for Section 2.5 1. If I :¼ ½a; b and I 0 :¼ ½a0 ; b0 are closed intervals in R, show that I I 0 if and only if a0 a and b b0 . 2. If S R is nonempty, show that S is bounded if and only if there exists a closed bounded interval I such that S I. 3. If S R is a nonempty bounded set, and I S :¼ ½inf S; sup S , show that S I S . Moreover, if J is any closed bounded interval containing S, show that I S J. 4. In the proof of Case (ii) of Theorem 2.5.1, explain why x, y exist in S. 5. Write out the details of the proof of Case (iv) in Theorem 2.5.1. 6. If I 1 I 2 I n is a nested sequence of intervals and if I n ¼ ½an ; bn , show that a1 a2 an and b1 b2 bn . 7. Let I n :¼ ½0; 1=n for n 2 N. Prove that T1 8. Let J n :¼ ð0; 1=nÞ for n 2 N. Prove that 9. Let K n :¼ ðn; 1Þ for n 2 N. Prove that n¼1 I n T1 ¼ f0g. n¼1 J n ¼ ;. n¼1 K n ¼ ;. T1 C02 12/09/2010 15:11:40 Page 53 53 2.5 INTERVALS 10. With the notation in the proofs of Theorems 2.5.2 and 2.5.3, show that we have h 2 T Also show that ½j; h ¼ 1 n¼1 I n . T1 n¼1 I n . 11. Show that the intervals obtained from the inequalities in (2) form a nested sequence. 12. Give the two binary representations of 3 8 and 7 16. 13. (a) Give the first four digits in the binary representation of 13. (b) Give the complete binary representation of 13. 14. Show that if ak ; bk 2 f0; 1; . . . ; 9g and if a1 a2 an b1 b2 bm þ þ þ n ¼ þ 2 þ þ m 6¼ 0; 10 102 10 10 10 10 then n ¼ m and ak ¼ bk for k ¼ 1; . . . ; n. 15. Find the decimal representation of 27. 16. Express 1 7 and 2 19 as periodic decimals. 17. What rationals are represented by the periodic decimals 1:25137 137 and 35:14653 653 ? C03 12/08/2010 11:35:1 Page 54 CHAPTER 3 SEQUENCES AND SERIES Now that the foundations of the real number system R have been laid, we are prepared to pursue questions of a more analytic nature, and we will begin with a study of the convergence of sequences. Some of the early results may be familiar to the reader from calculus, but the presentation here is intended to be rigorous and will lead to certain more profound theorems than are usually discussed in earlier courses. We will first introduce the meaning of the convergence of a sequence of real numbers and establish some basic, but useful, results about convergent sequences. We then present some deeper results concerning the convergence of sequences. These include the Monotone Convergence Theorem, the Bolzano-Weierstrass Theorem, and the Cauchy Criterion for convergence of sequences. It is important for the reader to learn both the theorems and how the theorems apply to special sequences. Because of the linear limitations inherent in a book it is necessary to decide where to locate the subject of infinite series. It would be reasonable to follow this chapter with a full discussion of infinite series, but this would delay the important topics of continuity, differentiation, and integration. Consequently, we have decided to compromise. A brief introduction to infinite series is given in Section 3.7 at the end of this chapter, and a more extensive treatment is given later in Chapter 9. Thus readers who want a fuller discussion of series at this point can move to Chapter 9 after completing this chapter. Augustin-Louis Cauchy Augustin-Louis Cauchy (1789–1857) was born in Paris just after the start of the French Revolution. His father was a lawyer in the Paris police department, and the family was forced to flee during the Reign of Terror. As a result, Cauchy’s early years were difficult and he developed strong anti-revolutionary and pro-royalist feelings. After returning to Paris, Cauchy’s father became secretary to the newly-formed Senate, which included the mathematicians Laplace and Lagrange. They were impressed by young Cauchy’s mathematical talent and helped him begin his career. # Bettmann/CORBIS He entered the Ecole Polytechnique in 1805 and soon established a reputation as an exceptional mathematician. In 1815, the year royalty was restored, he was appointed to the faculty of the Ecole Polytechnique, but his strong political views and his uncompromising standards in mathematics often resulted in bad relations with his colleagues. After the July revolution of 1830, Cauchy refused to sign the new loyalty oath and left France for eight years in self-imposed exile. In 1838, he accepted a minor teaching post in Paris, and in 1848 Napoleon III reinstated him to his former position at the Ecole Polytechnique, where he remained until his death. Cauchy was amazingly versatile and prolific, making substantial contributions to many areas, including real and complex analysis, number theory, differential equations, mathematical physics and probability. He published eight books and 789 papers, and his collected works fill 26 volumes. He was one of the most important mathematicians in the first half of the nineteenth century. 54 C03 12/08/2010 11:35:1 Page 55 3.1 SEQUENCES AND THEIR LIMITS 55 Section 3.1 Sequences and Their Limits A sequence in a set S is a function whose domain is the set N of natural numbers, and whose range is contained in the set S. In this chapter, we will be concerned with sequences in R and will discuss what we mean by the convergence of these sequences. 3.1.1 Definition A sequence of real numbers (or a sequence in R) is a function defined on the set N ¼ f1; 2; . . .g of natural numbers whose range is contained in the set R of real numbers. In other words, a sequence in R assigns to each natural number n ¼ 1, 2, . . . a uniquely determined real number. If X : N ! R is a sequence, we will usually denote the value of X at n by the symbol xn rather than using the function notation XðnÞ. The values xn are also called the terms or the elements of the sequence. We will denote this sequence by the notations X; ðxn Þ; ðxn : n 2 NÞ: Of course, we will often use other letters, such as Y ¼ ðyk Þ; Z ¼ ðzi Þ, and so on, to denote sequences. We purposely use parentheses to emphasize that the ordering induced by the natural order of N is a matter of importance. Thus, we distinguish notationally between the sequence ðxn : n 2 NÞ, whose infinitely many terms have an ordering, and the set of values fxn : n 2 Ng in the range of the sequence that are not ordered. For example, the sequence X :¼ ðð1Þn : n 2 N Þ has infinitely many terms that alternate between 1 and 1, whereas the set of values fð1Þn : n 2 Ng is equal to the set {1, 1}, which has only two elements. Sequences are often defined by giving a formula for the nth term xn. Frequently, it is convenient to list the terms of a sequence in order, stopping when the rule of formation seems evident. For example, we may define the sequence of reciprocals of the even numbers by writing 1 1 1 1 ; ; ; ; ; X :¼ 2 4 6 8 though a more satisfactory method is to specify the formula for the general term and write X :¼ 1 :n2N 2n or more simply X ¼ ð1=2nÞ. Another way of defining a sequence is to specify the value of x1 and give a formula for xnþ1 ðn 1Þ in terms of xn. More generally, we may specify x1 and give a formula for obtaining xnþ1 from x1 ; x2 ; . . . ; xn . Sequences defined in this manner are said to be inductively (or recursively) defined. 3.1.2 Examples (a) If b 2 R, the sequence B :¼ ðb; b; b; . . .Þ, all of whose terms equal b, is called the constant sequence b. Thus the constant sequence 1 is the sequence ð1; 1; 1; . . .Þ, and the constant sequence 0 is the sequence ð0; 0; 0; . . .Þ. C03 12/08/2010 11:35:2 56 Page 56 CHAPTER 3 SEQUENCES AND SERIES (b) If b 2 R, then B :¼ ðbn Þ is the sequence B ¼ ðb; b2 ; b3 ; . . . ; bn ; . . .Þ. In particular, if b ¼ 12, then we obtain the sequence 1 :n2N 2n 1 1 1 1 ; ; ;...; n;... : ¼ 2 4 8 2 (c) The sequence of ð2n : n 2 NÞ of even natural numbers can be defined inductively by x1 :¼ 2; xnþ1 :¼ xn þ 2; y1 :¼ 2; ynþ1 :¼ y1 þ yn : or by the definition (d) The celebrated Fibonacci sequence F :¼ ð f n Þ is given by the inductive definition f 1 :¼ 1; f 2 :¼ 1; f nþ1 :¼ f n1 þ f n ðn 2Þ: Thus each term past the second is the sum of its two immediate predecessors. The first ten terms of F are seen to be ð1; 1; 2; 3; 5; 8; 13; 21; 34; 55; . . .Þ. & The Limit of a Sequence There are a number of different limit concepts in real analysis. The notion of limit of a sequence is the most basic, and it will be the focus of this chapter. 3.1.3 Definition A sequence X ¼ ðxn Þ in R is said to converge to x 2 R, or x is said to be a limit of ðxn Þ, if for every e > 0 there exists a natural number KðeÞ such that for all n KðeÞ, the terms xn satisfy jxn xj < e. If a sequence has a limit, we say that the sequence is convergent; if it has no limit, we say that the sequence is divergent. Note The notation KðeÞ is used to emphasize that the choice of K depends on the value of e. However, it is often convenient to write K instead of KðeÞ. In most cases, a ‘‘small’’ value of e will usually require a ‘‘large’’ value of K to guarantee that the distance jxn xj between xn and x is less than e for all n K ¼ KðeÞ. When a sequence has limit x, we will use the notation lim X ¼ x or limðxn Þ ¼ x: We will sometimes use the symbolism xn ! x, which indicates the intuitive idea that the values xn ‘‘approach’’ the number x as n ! 1. 3.1.4 Uniqueness of Limits A sequence in R can have at most one limit. Proof. Suppose that x0 and x 00 are both limits of ðxn Þ. For each e > 0 there exist K0 such that jxn x0 j < e=2 for all n K 0 , and there exists K 00 such that jxn x 00 j < e=2 for all n K 00 . We let K be the larger of K0 and K 00 . Then for n K we apply the Triangle Inequality to get jx0 x 00 j ¼ jx0 xn þ xn x 00 j jx0 xn j þ jxn x 00 j < e=2 þ e=2 ¼ e: Since e > 0 is an arbitrary positive number, we conclude that x0 x 00 ¼ 0. Q.E.D. C03 12/08/2010 11:35:3 Page 57 3.1 SEQUENCES AND THEIR LIMITS 57 For x 2 R and e > 0, recall that the e-neighborhood of x is the set V e ðxÞ :¼ fu 2 R : ju xj < eg: (See Section 2.2.) Since u 2 V e ðxÞ is equivalent to ju xj < e, the definition of convergence of a sequence can be formulated in terms of neighborhoods. We give several different ways of saying that a sequence xn converges to x in the following theorem. 3.1.5 Theorem Let X ¼ (xn) be a sequence of real numbers, and let x 2 R. The following statements are equivalent. (a) X converges to x. (b) For every e > 0, there exists a natural number K such that for all n K, the terms xn satisfy jxn xj < e. (c) For every e > 0, there exists a natural number K such that for all n K, the terms xn satisfy x e < xn < x þ e. (d) For every e-neighborhood V e ðxÞ of x, there exists a natural number K such that for all n K, the terms xn belong to V e ðxÞ. Proof. The equivalence of (a) and (b) is just the definition. The equivalence of (b), (c), and (d) follows from the following implications: ju xj < e () e < u x < e () x e < u < x þ e () u 2 V e ðxÞ: Q.E.D. With the language of neighborhoods, one can describe the convergence of the sequence X ¼ ðxn Þ to the number x by saying: for each e-neighborhood Ve(x) of x, all but a finite number of terms of X belong to Ve (x). The finite number of terms that may not belong to the e-neighborhood are the terms x1 ; x2 ; . . . ; xK1 . Remark The definition of the limit of a sequence of real numbers is used to verify that a proposed value x is indeed the limit. It does not provide a means for initially determining what that value of x might be. Later results will contribute to this end, but quite often it is necessary in practice to arrive at a conjectured value of the limit by direct calculation of a number of terms of the sequence. Computers can be helpful in this respect, but since they can calculate only a finite number of terms of a sequence, such computations do not in any way constitute a proof of the value of the limit. The following examples illustrate how the definition is applied to prove that a sequence has a particular limit. In each case, a positive e is given and we are required to find a K, depending on e, as required by the definition. 3.1.6 Examples (a) lim(l=n) ¼ 0. If e > 0 is given, then 1=e > 0. By the Archimedean Property 2.4.3, there is a natural number K ¼ KðeÞ such that 1=K < e. Then, if n K, we have 1=n 1=K < e. Consequently, if n K, then 1 0 ¼ 1 < e: n n Therefore, we can assert that the sequence (1=n) converges to 0. C03 12/08/2010 11:35:3 58 Page 58 CHAPTER 3 SEQUENCES AND SERIES (b) lim(l=(n2 þ 1)) ¼ 0. Let e > 0 be given. To find K, we first note that if n 2 N, then 1 1 1 < : n2 þ 1 n2 n Now choose K such that 1=K < e, as in (a) above. Then n K implies that 1=n < e, and therefore 1 1 1 n2 þ 1 0 ¼ n2 þ 1 < n < e: Hence, we have shown that the limit of the sequence is zero. 3n þ 2 (c) lim ¼ 3: nþ1 Given e > 0, we want to obtain the inequality 3n þ 2 ð1Þ n þ 1 3 < e when n is sufficiently large. We first simplify the expression on the left: 3n þ 2 3n þ 2 3n 3 1 ¼ 1 < 1: ¼ n þ 1 3 ¼ nþ1 n þ 1 n þ 1 n Now if the inequality 1=n < e is satisfied, then the inequality (1) holds. Thus if 1=K < e, then for any n K, we also have 1=n < e and hence (1) holds. Therefore the limit of the sequence is 3. pffiffiffiffiffiffiffiffiffiffiffi pffiffiffi (d) limð n þ 1 nÞ ¼ 0. pffiffiffiffiffiffiffiffiffiffiffi pffiffiffi We multiply and divide by n þ 1 þ n to get pffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffiffiffiffiffiffiffiffiffi pffiffiffi ð n þ 1 nÞð n þ 1 þ nÞ nþ1n pffiffiffiffiffiffiffiffiffiffiffi pffiffiffi ¼ pffiffiffiffiffiffiffiffiffiffiffi pffiffiffi nþ1þ n nþ1þ n 1 1 ¼ pffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi n nþ1þ n pffiffiffi For a given e > 0, we obtain 1= n < e if and only if 1=n < e2 or n > 1=e2. Thus if we take p ffiffiffiffiffiffiffiffiffiffiffi p ffiffi ffi 2 K > 1=e , then n þ 1 n < e for all n > K. (For example, if we are given e ¼ 1=10, then K > 100 is required.) (e) If 0 < b < 1, then limðbn Þ ¼ 0. We will use elementary properties of the natural logarithm function. If e > 0 is given, we see that bn < e () n ln b < ln e () n > ln e=ln b: (The last inequality is reversed because ln b < 0.) Thus if we choose K to be a number such that K > ln e=ln b, then we will have 0 < bn < e for all n K. Thus we have limðbn Þ ¼ 0. For example, if b ¼ .8, and if e ¼ .01 is given, then we would need K > & ln :01=ln :8 20:6377. Thus K ¼ 21 would be an appropriate choice for e ¼ .01. Remark The KðeÞ Game In the notion of convergence of a sequence, one way to keep in mind the connection between the e and the K is to think of it as a game called the KðeÞ Game. In this game, Player A asserts that a certain number x is the limit of a sequence (xn). Player B challenges this assertion by giving Player A a specific value for e > 0. Player A must respond to the challenge by coming up with a value of K such that jxn xj < e for all n > K. If Player A can always find a value of K that works, then he wins, and the sequence is convergent. However, if Player B can give a specific value of e > 0 for which Player A C03 12/08/2010 11:35:5 Page 59 3.1 SEQUENCES AND THEIR LIMITS 59 cannot respond adequately, then Player B wins, and we conclude that the sequence does not converge to x. In order to show that a sequence X ¼ ðxn Þ does not converge to the number x, it is enough to produce one number e0 > 0 such that no matter what natural number K is chosen, one can find a particular nK satisfying nK K such that jxnK xj e0 . (This will be discussed in more detail in Section 3.4.) 3.1.7 Example The sequence ð0; 2; 0; 2; . . . ; 0; 2; . . .Þ does not converge to the number 0. If Player A asserts that 0 is the limit of the sequence, he will lose the KðeÞ Game when Player B gives him a value of e < 2. To be definite, let Player B give Player A the value e0 ¼ 1. Then no matter what value Player A chooses for K, his response will not be adequate, for Player B will respond by selecting an even number n > K. Then the corresponding value is xn ¼ 2 so that jxn 0j ¼ 2 > 1 ¼ e0 . Thus the number 0 is not & the limit of the sequence. Tails of Sequences It is important to realize that the convergence (or divergence) of a sequence X ¼ ðxn Þ depends only on the ‘‘ultimate behavior’’ of the terms. By this we mean that if, for any natural number m, we drop the first m terms of the sequence, then the resulting sequence Xm converges if and only if the original sequence converges, and in this case, the limits are the same. We will state this formally after we introduce the idea of a ‘‘tail’’ of a sequence. 3.1.8 Definition If X ¼ ðx1 ; x2 ; . . . ; xn ; . . .Þ is a sequence of real numbers and if m is a given natural number, then the m-tail of X is the sequence X m :¼ ðxmþn : n 2 NÞ ¼ ðxmþ1 ; xmþ2 ; . . .Þ For example, the 3-tail of the sequence X ¼ ð2; 4; 6; 8; 10; . . . ; 2n; . . .Þ, is the sequence X 3 ¼ ð8; 10; 12; . . . ; 2n þ 6; . . .Þ. 3.1.9 Theorem Let X ¼ ðxn : n 2 NÞ be a sequence of real numbers and let m 2 N. Then the m-tail X m ¼ ðxmþn : n 2 NÞ of X converges if and only if X converges. In this case, lim X m ¼ lim X. Proof. We note that for any p 2 N, the pth term of Xm is the ( p þ m)th term of X. Similarly, if q > m, then the qth term of X is the ðq mÞth term of Xm. Assume X converges to x. Then given any e > 0, if the terms of X for n KðeÞ satisfy jxn xj < e, then the terms of Xm for k KðeÞ m satisfy jxk xj < e. Thus we can take K m ðeÞ ¼ KðeÞ m, so that Xm also converges to x. Conversely, if the terms of Xm for k K m ðeÞ satisfy jxk xj < e, then the terms of X for n KðeÞ þ m satisfy jxn xj < e. Thus we can take KðeÞ ¼ K m ðeÞ þ m. Therefore, X converges to x if and only if Xm converges to x. Q.E.D. We shall sometimes say that a sequence X ultimately has a certain property if some tail of X has this property. For example, we say that the sequence ð3; 4; 5; 5; 5; . . . ; 5; . . .Þ is ‘‘ultimately constant.’’ On the other hand, the sequence ð3; 5; 3; 5; . . . ; 3; 5; . . .Þ is not ultimately constant. The notion of convergence can be stated using this terminology: A sequence X converges to x if and only if the terms of X are ultimately in every e-neighborhood of x. Other instances of this ‘‘ultimate terminology’’ will be noted later. C03 12/08/2010 11:35:5 60 Page 60 CHAPTER 3 SEQUENCES AND SERIES Further Examples In establishing that a number x is the limit of a sequence (xn), we often try to simplify the difference jxn xj before considering an e > 0 and finding a KðeÞ as required by the definition of limit. This was done in some of the earlier examples. The next result is a more formal statement of this idea, and the examples that follow make use of this approach. 3.1.10 Theorem Let (xn) be a sequence of real numbers and let x 2 R. If (an) is a sequence of positive real numbers with limðan Þ ¼ 0 and if for some constant C > 0 and some m 2 N we have jxn xj Can for all n m; then it follows that limðxn Þ ¼ x. Proof. If e > 0 is given, then since limðan Þ ¼ 0, we know there exists K ¼ Kðe=CÞ such that n K implies an ¼ jan 0j < e=C: Therefore it follows that if both n K and n m, then jxn xj Can < Cðe=CÞ ¼ e: Since e > 0 is arbitrary, we conclude that x ¼ limðxn Þ. Q.E.D. 1 3.1.11 Examples (a) If a > 0, then lim ¼ 0. 1 þ na Since a > 0, then 0 < na < 1 þ na, and therefore 0 < 1=(1 þ na) < 1=(na). Thus we have 1 1 1 for all n 2 N: 0 1 þ na a n Since limð1=nÞ ¼ 0, we may invoke Theorem 3.1.10 with C ¼ 1=a and m ¼ 1 to infer that lim ð1=ð1 þ naÞÞ ¼ 0. (b) If 0 < b < 1, then limðbn Þ ¼ 0. This limit was obtained earlier in Example 3.1.6(e). We will give a second proof that illustrates the use of Bernoulli’s Inequality (see Example 2.1.13(c)). Since 0 < b < 1, we can write b ¼ 1=(1 þ a), where a :¼ (1=b) 1 so that a > 0. By Bernoulli’s Inequality, we have ð1 þ aÞn 1 þ na. Hence 0 < bn ¼ 1 1 1 < : ð1 þ aÞn 1 þ na na Thus from Theorem 3.1.10 we conclude that limðbn Þ ¼ 0. In particular, if b ¼ .8, so that a ¼ .25, and if we are given e ¼ .01, then the preceding inequality gives us KðeÞ ¼ 4=ð:01Þ ¼ 400. Comparing with Example 3.1.6(e), where we obtained K ¼ 21, we see this method of estimation does not give us the ‘‘best’’ value of K. However, for the purpose of establishing the limit, the size of K is immaterial. (c) If c > 0, then limðc1=n Þ ¼ 1. The case c ¼ 1 is trivial, since then (c1/n) is the constant sequence ð1; 1; . . .Þ, which evidently converges to 1. C03 12/08/2010 11:35:6 Page 61 3.1 SEQUENCES AND THEIR LIMITS 61 If c > 1, then c1/n ¼ 1 þ dn for some dn > 0. Hence by Bernoulli’s Inequality 2.1.13(c), c ¼ ð1 þ d n Þn 1 þ nd n for n 2 N: Therefore we have c 1 nd n , so that d n ðc 1Þ=n. Consequently we have jc1=n 1j ¼ d n ðc 1Þ 1 n for n 2 N: We now invoke Theorem 3.1.10 to infer that limðc1=n Þ ¼ 1 when c > 1. Now suppose that 0 < c < 1; then c1/n ¼ 1=(1 þ hn) for some hn > 0. Hence Bernoulli’s Inequality implies that c¼ 1 1 1 < ; ð1 þ hn Þn 1 þ nhn nhn from which it follows that 0 < hn < 1=nc for n 2 N. Therefore we have 0 < 1 c1=n ¼ so that jc 1=n hn 1 < hn < nc 1 þ hn 1 1 1j < c n for n 2 N: We now apply Theorem 3.1.10 to infer that limðc1=n Þ ¼ 1 when 0 < c < 1. (d) limðn1=n Þ ¼ 1 Since n1=n > 1 for n > 1, we can write n1=n ¼ 1 þ kn for some kn > 0 when n > 1. Hence n ¼ ð1 þ kn Þn for n > 1. By the Binomial Theorem, if n > 1 we have n ¼ 1 þ nkn þ 12 nðn 1Þk2n þ 1 þ 12 nðn 1Þk2n ; whence it follows that n 1 12 nðn 1Þk2n : Hence k2n 2=n for n > 1. If e > 0 is given, it follows from the Archimedean Property that there exists a natural number Ne such that 2=Ne < e2. It follows that if n supf2; N e g then 2=n < e2, whence 0 < n1=n 1 ¼ kn ð2=nÞ1=2 < e: Since e > 0 is arbitrary, we deduce that limðn1=n Þ ¼ 1. & Exercises for Section 3.1 1. The sequence (xn) is defined by the following formulas for the nth term. Write the first five terms in each case: (a) xn :¼ 1 þ ð1Þn ; (b) xn :¼ ð1Þn =n; (c) xn :¼ 1 ; nðn þ 1Þ (d) x :¼ n2 1 : þ2 C03 12/08/2010 11:35:7 62 Page 62 CHAPTER 3 SEQUENCES AND SERIES 2. The first few terms of a sequence (xn) are given below. Assuming that the ‘‘natural pattern’’ indicated by these terms persists, give a formula for the nth term xn. (a) 5, 7, 9, 11, . . . , (b) 1=2, –1=4, 1=8, –1=16, . . . , (c) 1=2, 2=3, 3=4, 4=5, . . . , (d) 1, 4, 9, 16, . . . . 3. List (a) (b) (c) (d) the first five terms of the following inductively defined sequences. x1 :¼ 1; xnþ1 :¼ 3xn þ 1; y1 :¼ 2; ynþ1 :¼ 12 ðyn þ 2=yn Þ; z1 :¼ 1; z2 :¼ 2; znþ2 :¼ ðznþ1 þ zn Þ=ðznþ1 zn Þ; s1 :¼ 3; s2 :¼ 5; snþ2 :¼ sn þ snþ1 : 4. For any b 2 R, prove that limðb=nÞ ¼ 0. 5. Use the definition of the limit of a sequence to establish the following limits. n 2n ¼ 0; (b) lim ¼ 2; (a) lim 2 n þ1 nþ1 (c) lim 3n þ 1 3 ¼ ; 2n þ 5 2 6. Show that 1 (a) lim pffiffiffiffiffiffiffiffiffiffiffi ¼ 0; nþ7 (c) lim pffiffiffi n ¼ 0; nþ1 (d) 2 n 1 1 lim ¼ : 2n2 þ 3 2 (b) 2n ¼ 2; lim nþ2 (d) ð1Þn n ¼ 0: lim 2 n þ1 7. Let xn :¼ 1=lnðn þ 1Þ for n 2 N. (a) Use the definition of limit to show that limðxn Þ ¼ 0. (b) Find a specific value of K(e) as required in the definition of limit for each of (i) e ¼ 1=2, and (ii) e ¼ 1=10. 8. Prove that limðxn Þ ¼ 0 if and only if limðjxn jÞ ¼ 0. Give an example to show that the convergence of ðjxn jÞ need not imply the convergence of ðxn Þ. pffiffiffiffiffi 9. Show that if xn 0 for all n 2 N and limðxn Þ ¼ 0, then limð xn Þ ¼ 0. 10. Prove that if limðxn Þ ¼ x and if x > 0, then there exists a natural number M such that xn > 0 for all n M. 1 1 ¼ 0: 11. Show that lim n nþ1 pffiffiffiffiffiffiffiffiffiffiffiffiffi 12. Show that limð n2 þ 1 nÞ ¼ 0: 13. Show that limð1=3n Þ ¼ 0. 14. Let b 2 R satisfy 0 < b < 1. Show that limðnbn Þ ¼ 0. [Hint: Use the Binomial Theorem as in Example 3.1.11(d).] 15. Show that lim ð2nÞ1=n ¼ 1. 16. Show that limðn2 =n!Þ ¼ 0. 17. Show that limð2n =n!Þ ¼ 0. [Hint: If n 3, then 0 < 2n =n! 2 23 n2 .] 18. If limðxn Þ ¼ x > 0, show that there exists a natural number K such that if n K, then 1 2 x < xn < 2x. C03 12/08/2010 11:35:7 Page 63 3.2 LIMIT THEOREMS 63 Section 3.2 Limit Theorems In this section we will obtain some results that enable us to evaluate the limits of certain sequences of real numbers. These results will expand our collection of convergent sequences rather extensively. We begin by establishing an important property of convergent sequences that will be needed in this and later sections. 3.2.1 Definition A sequence X ¼ ðxn Þ of real numbers is said to be bounded if there exists a real number M > 0 such that jxn j M for all n 2 N. Thus, the sequence (xn) is bounded if and only if the set fxn : n 2 Ng of its values is a bounded subset of R. 3.2.2 Theorem A convergent sequence of real numbers is bounded. Proof. Suppose that limðxn Þ ¼ x and let e :¼ 1. Then there exists a natural number K ¼ K(1) such that jxn xj < 1 for all n K. If we apply the Triangle Inequality with n K we obtain jxn j ¼ jxn x þ xj jxn xj þ jxj < 1 þ jxj: If we set M :¼ sup jx1 j; jx2 j; . . . ; jxK1 j; 1 þ jxj ; then it follows that jxn j M for all n 2 N. Q.E.D. Remark We can also prove a convergent sequence (xn) is bounded using the language of neighborhoods. If Ve(x) is a given neighborhood of the limit x, then all but a finite number of terms of the sequence belong to Ve(x). Therefore, since Ve(x) is clearly bounded and finite sets are bounded, it follows that the sequence is bounded. We will now examine how the limit process interacts with the operations of addition, subtraction, multiplication, and division of sequences. If X ¼ (xn) and Y ¼ (yn) are sequences of real numbers, then we define their sum to be the sequence X þ Y : ¼ (xn þ yn), their difference to be the sequence X Y :¼ ðxn yn Þ, and their product to be the sequence X Y :¼ (xnyn). If c 2 R, we define the multiple of X by c to be the sequence cX :¼ (cxn). Finally, if Z ¼ (zn) is a sequence of real numbers with zn 6¼ 0 for all n 2 N, then we define the quotient of X and Z to be the sequence X=Z :¼ ðxn =zn Þ. For example, if X and Y are the sequences 1 1 1 1 ; ; ;...; ;... ; X :¼ ð2; 4; 6; . . . ; 2n; . . .Þ; Y :¼ 1 2 3 n then we have 3 9 19 2n2 þ 1 XþY ¼ ; ; ;...; ; ; 1 2 3 n 1 7 17 2n2 1 ; ; ;...; ; ; XY ¼ 1 2 3 n X Y ¼ ð2; 2; 2; . . . ; 2; . . .Þ; 3X ¼ ð6; 12; 18; . . . ; 6n; . . .Þ; X=Y ¼ ð2; 8; 18; . . . ; 2n2 ; . . .Þ: C03 12/08/2010 11:35:9 64 Page 64 CHAPTER 3 SEQUENCES AND SERIES We note that if Z is the sequence Z :¼ 0; 2; 0; . . . ; 1 þ ð1Þn ; . . . ; then we can define X þ Z, X – Z and X Z, but X=Z is not defined since some of the terms of Z are zero. We now show that sequences obtained by applying these operations to convergent sequences give rise to new sequences whose limits can be predicted. 3.2.3 Theorem (a) Let X ¼ (xn) and Y ¼ (yn) be sequences of real numbers that converge to x and y, respectively, and let c 2 R. Then the sequences X þ Y, X – Y, X Y, and cX converge to x þ y, x – y, xy, and cx, respectively. (b) If X ¼ (xn) converges to x and Z ¼ (zn) is a sequence of nonzero real numbers that converges to z and if z 6¼ 0, then the quotient sequence X=Z converges to x=z. Proof. (a) To show that limðxn þ yn Þ ¼ x þ y, we need to estimate the magnitude of jðxn þ yn Þ ðx þ yÞj. To do this we use the Triangle Inequality 2.2.3 to obtain jðxn þ yn Þ ðx þ yÞj ¼ jðxn xÞ þ ðyn yÞj jxn x j þ j yn yj: By hypothesis, if e > 0 there exists a natural number K1 such that if n K 1 , then jxn xj < e=2; also there exists a natural number K2 such that if n K 2 , then jyn yj < e=2. Hence if KðeÞ :¼ supfK 1 ; K 2 g, it follows that if n KðeÞ then jðxn þ yn Þ ðx þ yÞj jxn xj þ jyn yj 1 1 < e þ e ¼ e: 2 2 Since e > 0 is arbitrary, we infer that X þ Y ¼ ðxn þ yn Þ converges to x þ y. Precisely the same argument can be used to show that X Y ¼ ðxn yn Þ converges to x y. To show that X Y ¼ ðxn yn Þ converges to xy, we make the estimate jxn yn xyj ¼ jðxn yn xn yÞ þ ðxn y xyÞj jxn ðyn yÞj þ jðxn xÞyj ¼ jxn jjyn yj þ jxn xjjyj: According to Theorem 3.2.2 there exists a real number M 1 > 0 such that jxn j M 1 for all n 2 N and we set M :¼ supfM 1 ; jyjg. Hence we have jxn yn xyj Mjyn yj þ Mjxn xj: From the convergence of X and Y we conclude that if e > 0 is given, then there exist natural numbers K1 and K2 such that if n K 1 then jxn xj < e=2M, and if n K 2 then jyn yj < e=2M. Now let KðeÞ ¼ supfK 1 ; K 2 g; then, if n KðeÞ we infer that jxn yn xyj Mjyn yj þ Mjxn xj < Mðe=2MÞ þ Mðe=2MÞ ¼ e: Since e > 0 is arbitrary, this proves that the sequence X Y ¼ ðxn yn Þ converges to xy. The fact that cX ¼ (cxn) converges to cx can be proved in the same way; it can also be deduced by taking Y to be the constant sequence ðc; c; c; . . .Þ. We leave the details to the reader. C03 12/08/2010 11:35:10 Page 65 3.2 LIMIT THEOREMS 65 (b) We next show that if Z ¼ ðzn Þ is a sequence of nonzero numbers that converges to a nonzero limit z, then the sequence (l=zn) of reciprocals converges to 1=z. First let a :¼ 12 jzj so that a > 0. Since limðzn Þ ¼ z, there exists a natural number K1 such that if n K 1 then jzn zj < a. It follows from Corollary 2.2.4(a) of the Triangle Inequality that a jzn zj jzn j jzj for n K 1, whence it follows that 12 jzj ¼ jzj a jzn j for n K 1. Therefore 1=jzn j 2=jzj for n K 1 so we have the estimate 1 1 ¼ z zn ¼ 1 jz zn j z z z z jz zj n n n 2 jz zn j for all n K 1 : jzj2 Now, if e > 0 is given, there exists a natural number K2 such that if n K 2 then jzn zj < 12 ejzj2 . Therefore, it follows that if KðeÞ ¼ supfK 1 ; K 2 g, then 1 1 < e for all n > KðeÞ: z z n Since e > 0 is arbitrary, it follows that 1 1 lim ¼ : zn z The proof of (b) is now completed by taking Y to be the sequence ð1=zn Þ and using the fact that X Y ¼ ðxn =zn Þ converges to xð1=zÞ ¼ x=z. Q.E.D. Some of the results of Theorem 3.2.3 can be extended, by Mathematical Induction, to a finite number of convergent sequences. For example, if A ¼ ðan Þ; B ¼ ðbn Þ; . . . ; Z ¼ ðzn Þ are convergent sequences of real numbers, then their sum A þ B þ þ Z ¼ ðan þ bn þ þ zn Þ is a convergent sequence and (1) limðan þ bn þ þ zn Þ ¼ limðan Þ þ limðbn Þ þ þ limðzn Þ: Also their product A B Z :¼ ðan bn zn Þ is a convergent sequence and (2) limðan bn zn Þ ¼ ðlimðan ÞÞ ðlimðbn ÞÞ ðlimðzn ÞÞ: Hence, if k 2 N and if A ¼ ðan Þ is a convergent sequence, then (3) limðakn Þ ¼ ðlimðan ÞÞk : We leave the proofs of these assertions to the reader. 3.2.4 Theorem If X ¼ (xn) is a convergent sequence of real numbers and if xn 0 for all n 2 N, then x ¼ limðxn Þ 0. Proof. Suppose the conclusion is not true and that x < 0; then e :¼ x is positive. Since X converges to x, there is a natural number K such that x e < xn < x þ e for all n K: In particular, we have xK < x þ e ¼ x þ ðxÞ ¼ 0. But this contradicts the hypothesis that xn 0 for all n 2 N. Therefore, this contradiction implies that x 0. Q.E.D. We now give a useful result that is formally stronger than Theorem 3.2.4. 3.2.5 Theorem If X ¼ ðxn Þ and Y ¼ ðyn Þ are convergent sequences of real numbers and if xn yn for all n 2 N, then limðxn Þ limðyn Þ. C03 12/08/2010 11:35:10 66 Page 66 CHAPTER 3 SEQUENCES AND SERIES Proof. Let zn :¼ yn xn so that Z :¼ ðzn Þ ¼ Y X and zn 0 for all n 2 N. It follows from Theorems 3.2.3 and 3.2.4 that 0 lim Z ¼ limðyn Þ limðxn Þ; so that limðxn Þ limðyn Þ: Q.E.D. The next result asserts that if all the terms of a convergent sequence satisfy an inequality of the form a xn b, then the limit of the sequence satisfies the same inequality. Thus if the sequence is convergent, one may ‘‘pass to the limit’’ in an inequality of this type. 3.2.6 Theorem If X ¼ ðxn Þ is a convergent sequence and if a xn b for all n 2 N, then a limðxn Þ b. Proof. Let Y be the constant sequence ðb; b; b; . . .Þ. Theorem 3.2.5 implies that lim X lim Y ¼ b. Similarly one shows that a lim X. Q.E.D. The next result asserts that if a sequence Y is squeezed between two sequences that converge to the same limit, then it must also converge to this limit. 3.2.7 Squeeze Theorem Suppose that X ¼ ðxn Þ; Y ¼ ðyn Þ, and Z ¼ ðzn Þ are sequences of real numbers such that xn yn zn for all n 2 N; and that limðxn Þ ¼ limðzn Þ. Then Y ¼ ðyn Þ is convergent and limðxn Þ ¼ limðyn Þ ¼ limðzn Þ: Proof. Let w :¼ limðxn Þ ¼ limðzn Þ. If e > 0 is given, then it follows from the convergence of X and Z to w that there exists a natural number K such that if n K then jxn wj < e and jzn wj < e: Since the hypothesis implies that xn w yn w zn w for all n 2 N; it follows (why?) that e < yn w < e for all n K. Since e > 0 is arbitrary, this implies that limðyn Þ ¼ w. Q.E.D. Remark Since any tail of a convergent sequence has the same limit, the hypotheses of Theorems 3.2.4, 3.2.5, 3.2.6, and 3.2.7 can be weakened to apply to the tail of a sequence. For example, in Theorem 3.2.4, if X ¼ ðxn Þ is ‘‘ultimately positive’’ in the sense that there exists m 2 N such that xn 0 for all n m, then the same conclusion that x 0 will hold. Similar modifications are valid for the other theorems, as the reader should verify. 3.2.8 Examples (a) The sequence ðnÞ is divergent. It follows from Theorem 3.2.2 that if the sequence X :¼ ðnÞ is convergent, then there exists a real number M > 0 such that n ¼ jnj < M for all n 2 N. But this violates the Archimedean Property 2.4.3. C03 12/08/2010 11:35:11 Page 67 3.2 LIMIT THEOREMS 67 (b) The sequence ðð1Þn Þ is divergent. This sequence X ¼ ðð1Þn Þ is bounded (take M :¼ 1), so we cannot invoke Theorem 3.2.2. However, assume that a :¼ lim X exists. Let e :¼ 1 so that there exists a natural number K1 such that jð1Þn aj < 1 for all n K1: If n is an odd natural number with n K 1 this gives j 1 aj < 1, so that 2 < a < 0. On the other hand, if n is an even natural number with n K 1 , this inequality gives j1 aj < 1 so that 0 < a < 2. Since a cannot satisfy both of these inequalities, the hypothesis that X is convergent leads to a contradiction. Therefore the sequence X is divergent. 2n þ 1 (c) lim ¼ 2: n If we let X :¼ (2) and Y :¼ (1=n), then ðð2n þ 1Þ=nÞ ¼ X þ Y. Hence it follows from Theorem 3.2.3(a) that limðX þ YÞ ¼ lim X þ lim Y ¼ 2 þ 0 ¼ 2: 2n þ 1 (d) lim ¼ 2: nþ5 Since the sequences ð2n þ 1Þ and (n þ 5) are not convergent (why?), it is not possible to use Theorem 3.2.3(b) directly. However, if we write 2n þ 1 2 þ 1=n ¼ ; nþ5 1 þ 5=n we can obtain the given sequence as one to which Theorem 3.2.3(b) applies when we take X :¼ ð2 þ 1=nÞ and Z :¼ ð1 þ 5=nÞ. (Check that all hypotheses are satisfied.) Since lim X ¼ 2 and lim Z ¼ 1 6¼ 0, we deduce that limðð2n þ 1Þ=ðn þ 5ÞÞ ¼ 2=1 ¼ 2. 2n (e) lim 2 ¼ 0: n þ1 Theorem 3.2.3(b) does not apply directly. (Why?) We note that 2n 2 ; ¼ n2 þ 1 n þ 1=n but Theorem 3.2.3(b) does not apply here either, because ðn þ 1=nÞ is not a convergent sequence. (Why not?) However, if we write 2n 2=n ¼ ; n2 þ 1 1 þ 1=n2 then we can apply Theorem 3.2.3(b), since limð2=nÞ ¼ 0 and limð1 þ 1=n2 Þ ¼ 1 6¼ 0. Therefore limð2n=ðn2 þ 1ÞÞ ¼ 0=1 ¼ 0. sin n (f) lim ¼ 0: n We cannot apply Theorem 3.2.3(b) directly, since the sequence (n) is not convergent [neither is the sequence (sin n)]. It does not appear that a simple algebraic manipulation will enable us to reduce the sequence into one to which Theorem 3.2.3 will apply. However, if we note that 1 sin n 1, then it follows that 1 sin n 1 n n n for all n 2 N: C03 12/08/2010 11:35:11 68 Page 68 CHAPTER 3 SEQUENCES AND SERIES Hence we can apply the Squeeze Theorem 3.2.7 to infer that limðn1 sin nÞ ¼ 0. (We note that Theorem 3.1.10 could also be applied to this sequence.) (g) Let X ¼ ðxn Þ be a sequence of real numbers that converges to x 2 R. Let p be a polynomial; for example, let pðtÞ :¼ ak tk þ ak1 tk1 þ þ a1 t þ a0 ; where k 2 N and aj 2 R for j ¼ 0; 1; . . . ; k. It follows from Theorem 3.2.3 that the sequence ð pðxn ÞÞ converges to pðxÞ. We leave the details to the reader as an exercise. (h) Let X ¼ ðxn Þ be a sequence of real numbers that converges to x 2 R. Let r be a rational function (that is, rðtÞ :¼ pðtÞ=qðtÞ, where p and q are polynomials). Suppose that qðxn Þ 6¼ 0 for all n 2 N and that qðxÞ 6¼ 0. Then the sequence ðrðxn ÞÞ converges to & rðxÞ ¼ pðxÞ=qðxÞ. We leave the details to the reader as an exercise. We conclude this section with several results that will be useful in the work that follows. 3.2.9 Theorem Let the sequence X ¼ ðxn Þ converge to x. Then the sequence ðjxn jÞ of absolute values converges to jxj. That is, if x ¼ limðxn Þ, then jxj ¼ limðjxn jÞ. Proof. It follows from the Triangle Inequality (see Corollary 2.2.4(a)) that jxn j jxj jxn xj for all n 2 N: The convergence of ðjxn jÞ to jxj is then an immediate consequence of the convergence of ðxn Þ to x. Q.E.D. 3.2.10 Theorem Let X ¼ ðxn Þ be a sequence of real numbers that converges to x and pffiffiffiffiffi suppose thatpxffiffiffin 0. Then the sequence ð xn Þ of positive square roots converges and pffiffiffiffiffi limð xn Þ ¼ x. Proof. It follows from Theorem 3.2.4 that x ¼ limðxn Þ 0 so the assertion makes sense. We now consider the two cases: (i) x ¼ 0 and (ii) x > 0. Case (i) If x ¼ 0, let e > 0 be given. Since xn ! 0 there exists a natural number K such that if n K then 0 xn ¼ xn 0 < e2 : pffiffiffiffiffi Therefore [see Example 2.1.13(a)], 0 xn < e for n K. Since e > 0 is arbitrary, this pffiffiffiffiffi implies that xn ! 0. pffiffiffi Case (ii) If x > 0, then x > 0 and we note that pffiffiffiffiffi pffiffiffi pffiffiffiffiffi pffiffiffi pffiffiffiffiffi pffiffiffi ð xn xÞð xn þ xÞ xn x xn x ¼ ¼ pffiffiffiffiffi pffiffiffi pffiffiffiffiffi pffiffiffi xn þ x xn þ x pffiffiffiffiffi pffiffiffi pffiffiffi Since xn þ x x > 0, it follows that pffiffiffiffiffi pffiffiffi 1 xn x pffiffiffi jxn xj: x pffiffiffi pffiffiffiffiffi The convergence of xn ! x follows from the fact that xn ! x. Q.E.D. For certain types of sequences, the following result provides a quick and easy ‘‘ratio test’’ for convergence. Related results can be found in the exercises. C03 12/08/2010 11:35:12 Page 69 3.2 LIMIT THEOREMS 69 3.2.11 Theorem Let ðxn Þ be a sequence of positive real numbers such that L :¼ limðxnþ1 =xn Þ exists. If L < 1, then ðxn Þ converges and limðxn Þ ¼ 0. Proof. By 3.2.4 it follows that L 0. Let r be a number such that L < r < 1, and let e :¼ r L > 0. There exists a number K 2 N such that if n K then xnþ1 x L < e: n It follows from this (why?) that if n K, then xnþ1 < L þ e ¼ L þ ðr LÞ ¼ r: xn Therefore, if n K, we obtain 0 < xnþ1 < xn r < xn1 r2 < < xK r nKþ1 : If we set C :¼ xK =rK , we see that 0 < xnþ1 < Cr nþ1 for all n K. Since 0 < r < 1, it follows from 3.1.11(b) that limðrn Þ ¼ 0 and therefore from Theorem 3.1.10 that limðxn Þ ¼ 0. Q.E.D. As an illustration of the utility of the preceding theorem, consider the sequence ðxn Þ given by xn :¼ n=2n. We have xnþ1 n þ 1 2n 1 1 1þ ; ¼ nþ1 ¼ 2 n xn n 2 so that limðxnþ1 =xn Þ ¼ 12. Since 12 < 1, it follows from Theorem 3.2.11 that limðn=2n Þ ¼ 0. Exercises for Section 3.2 1. For xn given by the following formulas, establish either the convergence or the divergence of the sequence X ¼ ðxn Þ. n ð1Þn n (a) xn :¼ ; (b) xn :¼ ; nþ1 nþ1 n2 2n2 þ 3 ; (d) xn :¼ 2 : nþ1 n þ1 2. Give an example of two divergent sequences X and Y such that: (a) their sum X þ Y converges, (b) their product XY converges. (c) xn :¼ 3. Show that if X and Y are sequences such that X and X þ Y are convergent, then Y is convergent. 4. Show that if X and Y are sequences such that X converges to x 6¼ 0 and XY converges, then Y converges. 5. Show that the following sequences are not convergent. (a) ð2n Þ; (b) ð1Þn n2 : 6. Find the limits of the following sequences: (a) lim ð2 þ 1=nÞ2 ; pffiffiffi n1 (c) lim pffiffiffi ; nþ1 (b) (d) ð1Þn ; nþ2 nþ1 lim pffiffiffi : n n lim C03 12/08/2010 11:35:13 70 Page 70 CHAPTER 3 SEQUENCES AND SERIES 7. If ðbn Þ is a bounded sequence and limðan Þ ¼ 0, show that limðan bn Þ ¼ 0. Explain why Theorem 3.2.3 cannot be used. 8. Explain why the result in equation (3) before Theorem 3.2.4 cannot be used to evaluate the limit of the sequence ðð1 þ 1=nÞn Þ. pffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi 9. Let yn :¼ n þ 1 n for n 2 N. Show that ð nyn Þ converges. Find the limit. 10. Determine the limits of the following sequences. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (b) (a) ð 4 n2 þ n 2nÞ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð n2 þ 5n nÞ: 11. Determine following limits. the pffiffiffi (a) lim ð3 nÞ1=2n ; (b) lim ðn þ 1Þ1=lnðnþ1Þ . nþ1 a þ bnþ1 12. If 0 < a < b; determine lim : n a þ bn pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 13. If a > 0; b > 0; show that lim ðn þ aÞðn þ bÞ n ¼ ða þ bÞ=2. 14. Use the Squeeze Theorem 3.2.7 to determine the limits of the2 following, 2 (b) ðn!Þ1=n : (a) ðn1=n Þ; 15. Show that if zn :¼ ðan þ bn Þ1=n where 0 < a < b, then limðzn Þ ¼ b. 16. Apply Theorem 3.2.11 to the following sequences, where a, b satisfy 0 < a < 1; b > 1. (a) ðan Þ; (b) ðbn =2n Þ; n (c) ðn=b Þ; (d) ð23n =32n Þ: 17. (a) Give an example of a convergent sequence ðxn Þ of positive numbers with limðxnþ1 =xn Þ ¼ 1. (b) Give an example of a divergent sequence with this property. (Thus, this property cannot be used as a test for convergence.) 18. Let X ¼ (xn) be a sequence of positive real numbers such that limðxnþ1 =xn Þ ¼ L > 1. Show that X is not a bounded sequence and hence is not convergent. 19. Discuss the convergence of the following sequences, where a, b satisfy 0 < a < 1; b > 1. (b) ðbn =n2 Þ; (a) ðn2 an Þ; (c) ðbn =n!Þ; (d) ðn!=nn Þ: 20. Let ðxn Þ be a sequence of positive real numbers such that limðx1=n n Þ ¼ L < 1. Show that there exists a number r with 0 < r < 1 such that 0 < xn < rn for all sufficiently large n 2 N. Use this to show that limðxn Þ ¼ 0. 21. (a) Give an example of a convergent sequence (xn) of positive numbers with limðxn1=n Þ ¼ 1. (b) Give an example of a divergent sequence (xn) of positive numbers with limðxn1=n Þ ¼ 1. (Thus, this property cannot be used as a test for convergence.) 22. Suppose that ðxn Þ is a convergent sequence and ðyn Þ is such that for any e > 0 there exists M such that jxn yn j < e for all n M. Does it follow that (yn) is convergent? 23. Show that if (xn) and (yn) are convergent sequences, then the sequences (un) and (vn) defined by un :¼ maxfxn ; yn g and vn :¼ minfxn ; yn g are also convergent. (See Exercise 2.2.18.) 24. Show that if ðxn Þ; ðyn Þ; ðzn Þ are convergent sequences, then the sequence (wn) defined by wn :¼ midfxn ; yn ; zn g is also convergent. (See Exercise 2.2.19.) Section 3.3 Monotone Sequences Until now, we have obtained several methods of showing that a sequence X ¼ (xn) of real numbers is convergent: C03 12/08/2010 11:35:14 Page 71 3.3 MONOTONE SEQUENCES 71 (i) We can use Definition 3.1.3 or Theorem 3.1.5 directly. This is often (but not always) difficult to do. (ii) We can dominate jxn xj by a multiple of the terms in a sequence ðan Þ known to converge to 0, and employ Theorem 3.1.10. (iii) We can identify X as a sequence obtained from other sequences that are known to be convergent by taking tails, algebraic combinations, absolute values, or square roots, and employ Theorems 3.1.9, 3.2.3, 3.2.9, or 3.2.10. (iv) We can ‘‘squeeze’’ X between two sequences that converge to the same limit and use Theorem 3.2.7. (v) We can use the ‘‘ratio test’’ of Theorem 3.2.11. Except for (iii), all of these methods require that we already know (or at least suspect) the value of the limit, and we then verify that our suspicion is correct. There are many instances, however, in which there is no obvious candidate for the limit of a sequence, even though a preliminary analysis may suggest that convergence is likely. In this and the next two sections, we shall establish results that can be used to show a sequence is convergent even though the value of the limit is not known. The method we introduce in this section is more restricted in scope than the methods we give in the next two, but it is much easier to employ. It applies to sequences that are monotone in the following sense. 3.3.1 Definition Let X ¼ (xn) be a sequence of real numbers. We say that X is increasing if it satisfies the inequalities x1 x2 xn xnþ1 : We say that X is decreasing if it satisfies the inequalities x1 x2 xn xnþ1 : We say that X is monotone if it is either increasing or decreasing. The following sequences are increasing: ð1; 2; 3; 4; . . . ; n; . . .Þ; ð1; 2; 2; 3; 3; 3; . . .Þ; ða; a2 ; a3 ; . . . ; an ; . . .Þ if a > 1: The following sequences are decreasing: ð1; 1=2; 1=3; . . . ; 1=n; . . .Þ; ð1; 1=2; 1=22 ; . . . ; 1=2n1 ; . . .Þ; ðb; b2 ; b3 ; . . . ; bn ; . . .Þ if 0 < b < 1: The following sequences are not monotone: þ1; 1; þ1; . . . ; ð1Þnþ1 ; . . . ; 1; þ2; 3; . . . ; ð1Þn n . . . The following sequences are not monotone, but they are ‘‘ultimately’’ monotone: ð7; 6; 2; 1; 2; 3; 4; . . .Þ; ð2; 0; 1; 1=2; 1=3; 1=4; . . .Þ: 3.3.2 Monotone Convergence Theorem A monotone sequence of real numbers is convergent if and only if it is bounded. Further: C03 12/08/2010 11:35:14 72 Page 72 CHAPTER 3 SEQUENCES AND SERIES (a) If X ¼ (xn) is a bounded increasing sequence, then limðxn Þ ¼ supfxn : n 2 Ng: (b) If Y ¼ (yn) is a bounded decreasing sequence, then limðyn Þ ¼ inffyn : n 2 Ng: Proof. It was seen in Theorem 3.2.2 that a convergent sequence must be bounded. Conversely, let X be a bounded monotone sequence. Then X is either increasing or decreasing. (a) We first treat the case where X ¼ (xn) is a bounded, increasing sequence. Since X is bounded, there exists a real number M such that xn M for all n 2 N. According to the Completeness Property 2.3.6, the supremum x ¼ supfxn : n 2 Ng exists in R; we will show that x ¼ limðxn Þ. If e > 0 is given, then x e is not an upper bound of the set fxn : n 2 Ng, and hence there exists xK such that x e < xK . The fact that X is an increasing sequence implies that xK xn whenever n K, so that x e < xK xn x < x þ e for all n K: Therefore we have jxn x j < e for all n K: Since e > 0 is arbitrary, we conclude that (xn) converges to x. (b) If Y ¼ (yn) is a bounded decreasing sequence, then it is clear that X :¼ Y ¼ ðyn Þ is a bounded increasing sequence. It was shown in part (a) that lim X ¼ supfyn : n 2 Ng, Now lim X ¼ lim Y and also, by Exercise 2.4.4(b), we have supfyn : n 2 Ng ¼ inffyn : n 2 Ng: Therefore lim Y ¼ lim X ¼ inffyn : n 2 Ng: Q.E.D. The Monotone Convergence Theorem establishes the existence of the limit of a bounded monotone sequence. It also gives us a way of calculating the limit of the sequence provided we can evaluate the supremum in case (a), or the infimum in case (b). Sometimes it is difficult to evaluate this supremum (or infimum), but once we know that it exists, it is often possible to evaluate the limit by other methods. pffiffiffi 3.3.3 Examples (a) limð1= nÞ ¼ 0. It is possible to handle this sequence by using Theorem 3.2.10; however, we shallpuse ffiffiffiffiffiffi the Monotone Convergence Theorem. Clearly 0 is a lower bound for thepset ffiffiffi f1= n : n 2 Ng, and it is not pffiffiffi difficult to show that 0 is the infimum of the set f1= n : n 2 Ng; hence 0 ¼ limð1= nÞ. pffiffiffi On the other hand, once we know that X :¼ ð1= nÞ is bounded pffiffiffi and decreasing, we know that it converges to some real number x. Since X ¼ ð1= nÞ converges to x, it follows from Theorem 3.2.3 that X X ¼ ð1=nÞ converges to x2 . Therefore x2 ¼ 0, whence x ¼ 0. (b) Let hn :¼ 1 þ 1=2 þ 1=3 þ þ 1=n for n 2 N. Since hnþ1 ¼ hn þ 1=ðn þ 1Þ > hn , we see that ðhn Þ is an increasing sequence. By the Monotone Convergence Theorem 3.3.2, the question of whether the sequence is convergent or not is reduced to the question of whether the sequence is bounded or not. Attempts to use direct numerical calculations to arrive at a conjecture concerning the possible boundedness of the sequence ðhn Þ lead to inconclusive frustration. A computer run will reveal the approximate values hn 11:4 for n ¼ 50; 000, and hn 12:1 for n ¼ 100;000. Such C03 12/08/2010 11:35:15 Page 73 3.3 MONOTONE SEQUENCES 73 numerical facts may lead the casual observer to conclude that the sequence is bounded. However, the sequence is in fact divergent, which is established by noting that 1 1 1 1 1 þ h2 n ¼ 1 þ þ þ þ n1 þ þ n 2 3 4 2 2 þ1 1 1 1 1 1 þ > 1þ þ þ þ n þ þ n 2 4 4 2 2 1 1 1 ¼ 1 þ þ þ þ 2 2 2 n ¼ 1þ : 2 Since ðhn Þ is unbounded, Theorem 3.2.2 implies that it is divergent. (This proves that the infinite series known as the harmonic series diverges. See Example 3.7.6(b) in Section 3.7.) The terms hn increase extremely slowly. For example, it can be shown that to achieve hn > 50 would entail approximately 5.2 1021 additions, and a normal computer performing 400 million additions a second would require more than 400,000 years to perform the calculation (there are 31,536,000 seconds in a year). A supercomputer that can perform more than a trillion additions a second would take more than 164 years to reach that modest goal. And the IBM Roadrunner supercomputer at a speed of a quadrillion operations per & second would take over a year and a half. Sequences that are defined inductively must be treated differently. If such a sequence is known to converge, then the value of the limit can sometimes be determined by using the inductive relation. For example, suppose that convergence has been established for the sequence ðxn Þ defined by x1 ¼ 2; xnþ1 ¼ 2 þ 1 ; xn n 2 N: If we let x ¼ limðxn Þ, then we also have x ¼ limðxnþ1 Þ since the 1-tail ðxnþ1 Þ converges to the same limit. Further, we see that xn 2, so that x 6¼ 0 and xn 6¼ 0 for all n 2 N. Therefore, we may apply the limit theorems for sequences to obtain x ¼ limðxnþ1 Þ ¼ 2 þ 1 1 ¼2þ : limðxn Þ x Thus, the limit x is a solution of the quadratic equation x2 2x p ffiffi1ffi ¼ 0, and since x must be positive, we find that the limit of the sequence is x ¼ 1 þ 2. Of course, the issue of convergence must not be ignored or casually assumed. For example, if we assumed the sequence ðyn Þ defined by y1 :¼ 1; ynþ1 :¼ 2yn þ 1 is convergent with limit y, then we would obtain y ¼ 2y þ 1, so that y ¼ 1. Of course, this is absurd. In the following examples, we employ this method of evaluating limits, but only after carefully establishing convergence using the Monotone Convergence Theorem. Additional examples of this type will be given in Section 3.5. 3.3.4 Examples (a) Let Y ¼ (yn) be defined inductively by y1 :¼ 1; ynþ1 :¼ 14 ð2yn þ3Þ for n 1. We shall show that lim Y ¼ 3=2. C03 12/08/2010 11:35:15 74 Page 74 CHAPTER 3 SEQUENCES AND SERIES Direct calculation shows that y2 ¼ 5=4. Hence we have y1 < y2 < 2. We show, by Induction, that yn < 2 for all n 2 N. Indeed, this is true for n ¼ 1; 2. If yk < 2 holds for some k 2 N, then ykþ1 ¼ 14 ð2yk þ 3Þ < 14 ð4 þ 3Þ ¼ 74 < 2; so that ykþ1 < 2. Therefore yn < 2 for all n 2 N. We now show, by Induction, that yn < ynþ1 for all n 2 N. The truth of this assertion has been verified for n ¼ 1. Now suppose that yk < ykþ1 for some k; then 2yk þ 3 < 2ykþ1 þ 3, whence it follows that ykþ1 ¼ 14 ð2yk þ 3Þ < 14 ð2ykþ1 þ 3Þ ¼ ykþ2 : Thus yk < ykþ1 implies that ykþ1 < ykþ2 . Therefore yn < ynþ1 for all n 2 N. We have shown that the sequence Y ¼ ðyn Þ is increasing and bounded above by 2. It follows from the Monotone Convergence Theorem that Y converges to a limit that is at most 2. In this case it is not so easy to evaluate limðyn Þ by calculating supfyn : n 2 Ng. However, there is another way to evaluate its limit. Since ynþ1 ¼ 14 ð2yn þ 3Þ for all n 2 N, the nth term in the 1-tail Y1 of Y has a simple algebraic relation to the nth term of Y. Since, by Theorem 3.1.9, we have y :¼ lim Y 1 ¼ lim Y, it therefore follows from Theorem 3.2.3 (why?) that y ¼ 14 ð2y þ 3Þ; from which it follows that y ¼ 3=2. pffiffiffiffiffiffiffi (b) Let Z ¼ (zn) be the sequence of real numbers defined by z1 :¼ 1; znþ1 :¼ 2zn for n 2 N. We will show that limðzpn Þffiffiffi ¼ 2. Note that z1 ¼ 1 and z2 ¼ 2; hence 1 z1 < z2 < 2. We claim that the sequence Z is increasing and bounded above by 2. To show this we will show, by Induction, that 1 zn < znþ1 < 2 for all n 2 N. This fact has been verified for n ¼ 1. Suppose that it is true for n ¼ k; then 2 2zk < 2zkþ1 < 4, whence it follows (why?) that 1< pffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi 2 zkþ1 ¼ 2zk < zkþ2 ¼ 2zkþ1 < 4 ¼ 2: [In this last step we have used Example 2.1.13(a).] Hence the validity of the inequality 1 zk < zkþ1 < 2 implies the validity of 1 zkþ1 < zkþ2 < 2. Therefore 1 zn < znþ1 < 2 for all n 2 N. Since Z ¼ ðzn Þ is a bounded increasing sequence, it follows from the Monotone Convergence Theorem that it converges to a number z :¼ supfzn g. It may be shown directly that supfzn g ¼ 2, sop that ffiffiffiffiffiffizffi ¼ 2. Alternatively we may use the method employed in part (a). The relation znþ1 ¼ 2zn gives a relation between the nth term of the 1-tail Z1 of Z and the nth term of Z. By Theorem 3.1.9, we have lim Z 1 ¼ z ¼ lim Z. Moreover, by Theorems 3.2.3 and 3.2.10, it follows that the limit z must satisfy the relation pffiffiffiffiffi z ¼ 2z: Hence z must satisfy the equation z2 ¼ 2z, which has the roots z ¼ 0; 2. Since the terms of z ¼ ðzn Þ all satisfy 1 zn 2, it follows from Theorem 3.2.6 that we must have & 1 z 2. Therefore z ¼ 2. C03 12/08/2010 11:35:16 Page 75 3.3 MONOTONE SEQUENCES 75 The Calculation of Square Roots We now give an application of the Monotone Convergence Theorem to the calculation of square roots of positive numbers. 3.3.5 Example pffiffiffi Let a > 0; we will construct a sequence (sn) of real numbers that converges to a. Let s1 > 0 be arbitrary andpdefine snþ1 :¼ 12 ðsn þ a=sn Þ for n 2 N. We now show that ffiffiffi the sequence (sn) converges to a. (This process for calculating square roots was known in Mesopotamia before 1500 B.C.) We first show that s2n a for n 2. Since sn satisfies the quadratic equation 2 sn 2snþ1 sn þ a ¼ 0, this equation has a real root. Hence the discriminant 4s2nþ1 4a must be nonnegative; that is, s2nþ1 a for n 1. To see that (sn) is ultimately decreasing, we note that for n 2 we have 1 a 1 ðs2 aÞ sn þ 0: sn snþ1 ¼ sn ¼ n 2 sn 2 sn Hence, snþ1 sn for all n 2. The Monotone Convergence Theorem implies that s :¼ limðsn Þ exists. Moreover, from Theorem 3.2.3, the limit s must satisfy the relation s¼ 1 a sþ ; 2 s pffiffiffi whence it follows (why?) that s ¼ a=s or s2 ¼ a. Thus s ¼ a. For the purposes of calculation, an estimate of how rapidly pffiffiffi it is often important to phave ffiffiffi the sequence (sn) converges to a . As above, we have a s n for all n 2, whence it pffiffiffi follows that a=sn a sn . Thus we have pffiffiffi 0 sn a sn a=sn ¼ ðs2n aÞ=sn for n 2: pffiffiffi & Using this inequality we can calculate a to any desired degree of accuracy. Euler’s Number We conclude this section by introducing a sequence that converges to one of the most important ‘‘transcendental’’ numbers in mathematics, second in importance only to p. 3.3.6 Example Let en :¼ ð1 þ 1=nÞn for n 2 N. We will now show that the sequence E ¼ (en) is bounded and increasing; hence it is convergent. The limit of this sequence is the famous Euler number e, whose approximate value is 2:718 281 828 459 045 . . . ; which is taken as the base of the ‘‘natural’’ logarithm. If we apply the Binomial Theorem, we have 1 n n 1 nðn 1Þ 1 nðn 1Þðn 2Þ 1 2þ 3 en ¼ 1 þ ¼1þ þ n 1 n 2! n 3! n þ þ nðn 1Þ 2 1 1 n: n! n C03 12/08/2010 11:35:16 76 Page 76 CHAPTER 3 SEQUENCES AND SERIES If we divide the powers of n into the terms in the numerators of the binomial coefficients, we get 1 1 1 1 2 1 1 en ¼ 1 þ 1 þ þ 1 2! n 3! n n 1 1 2 n1 1 þ þ 1 1 : n! n n n Similarly we have enþ1 1 1 1 1 2 1 1 ¼1þ1 þ þ 1 2! nþ1 3! nþ1 nþ1 1 1 2 n1 1 þ þ 1 1 n! nþ1 nþ1 nþ1 1 1 2 n 1 þ 1 1 : ðn þ 1Þ! nþ1 nþ1 nþ1 Note that the expression for en contains n þ 1 terms, while that for enþ1 contains n þ 2 terms. Moreover, each term appearing in en is less than or equal to the corresponding term in enþ1 , and enþ1 has one more positive term. Therefore we have 2 e1 < e2 < < en < enþ1 < ; so that the terms of E are increasing. To show that the terms of E are bounded above, we note that if p ¼ 1; 2; . . . ; n, then ð1 p=nÞ < 1. Moreover 2p1 p! [see 1.2.4(e)] so that 1=p! 1=2p1 . Therefore, if n > 1, then we have 1 1 1 2 < en < 1 þ 1 þ þ 2 þ þ n1 : 2 2 2 Since it can be verified that [see 1.2.4(f)] 1 1 1 1 þ þ þ n1 ¼ 1 n1 < 1; 2 22 2 2 we deduce that 2 < en < 3 for all n 2 N. The Monotone Convergence Theorem implies that the sequence E converges to a real number that is between 2 and 3. We define the number e to be the limit of this sequence. By refining our estimates we can find closer rational approximations to e, but we cannot evaluate it exactly, since e is an irrational number. However, it is possible to calculate e to as many decimal places as desired. The reader should use a calculator (or a & computer) to evaluate en for ‘‘large’’ values of n. Leonhard Euler Leonhard Euler (1707–1783) was born near Basel, Switzerland. His clergyman father hoped his son would follow him into the ministry, but when Euler entered the University of Basel at age 14, he studied medicine, physics, astronomy, and mathematics as well as theology. His mathematical talent was noticed by Johann Bernoulli, who became his mentor. In 1727, Euler traveled to Russia to join Bernoulli’s son, Daniel, at the new St. Petersburg Academy. There he met and married Katharina Gsell, the daughter of a Swiss artist. During their 40-year marriage, they had 13 children, but only five survived childhood. C03 12/08/2010 11:35:17 Page 77 3.3 MONOTONE SEQUENCES 77 In 1741, Euler accepted an offer from Frederick the Great to join the Berlin Academy, where he stayed for 25 years. During this period he wrote landmark books on a relatively new subject called calculus and a steady stream of papers on mathematics and science. In response to a request for instruction in science from the Princess of Anhalt-Dessau, he wrote her nearly 200 letters on science that later became famous in a book titled Letters to a German Princess. When Euler lost vision in one eye, Frederick thereafter referred to him as his mathematical ‘‘cyclops.’’ In 1766, he happily returned to Russia at the invitation of Catherine the Great. His eyesight continued to deteriorate and in 1771 he became totally blind following an eye operation. Incredibly, his blindness made little impact on his mathematics output, for he wrote several books and over 400 papers while blind. He remained active until the day of his death. Euler’s productivity was remarkable. He wrote textbooks on physics, algebra, calculus, real and complex analysis, and differential geometry. He also wrote hundreds of papers, many winning prizes. A current edition of his collected works consists of 74 volumes. Exercises for Section 3.3 1. Let x1 :¼ 8 and xnþ1 :¼ 12 xn þ 2 for n 2 N. Show that ðxn Þ is bounded and monotone. Find the limit. 2. Let x1 > 1 and xnþ1 :¼ 2 1=xn for n 2 N. show that ðxn Þ is bounded and monotone. Find the limit. pffiffiffiffiffiffiffiffiffiffiffiffiffi 3. Let x1 2 and xnþ1 :¼ 1 þ xn 1 for n 2 N. Show that ðxn Þ is decreasing and bounded below by 2. Find the limit. pffiffiffiffiffiffiffiffiffiffiffiffiffi 4. Let x1 :¼ 1 and xnþ1 :¼ 2 þ xn for n 2 N. Show that ðxn Þ converges and find the limit. pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 5. Let y1 :¼ p, where p > 0, and ynþ1 :¼ p þ yn for n 2 N. Show that ðyn Þ converges and find pffiffiffi the limit. [Hint: One upper bound is 1 þ 2 p.] pffiffiffiffiffiffiffiffiffiffiffiffiffi 6. Let a > 0 and let z1 > 0: Define znþ1 :¼ a þ zn for n 2 N. Show that ðzn Þ converges and find the limit. 7. Let x1 :¼ a > 0 and xnþ1 :¼ xn þ 1=xn for n 2 N. Determine whether ðxn Þ converges or diverges. 8. Let ðan Þ be an increasing sequence, ðbn Þ be a decreasing sequence, and assume that an bn for all n 2 N. Show that limðan Þ limðbn Þ, and thereby deduce the Nested Intervals Property 2.5.2 from the Monotone Convergence Theorem 3.3.2. 9. Let A be an infinite subset of R that is bounded above and let u :¼ sup A. Show there exists an increasing sequence ðxn Þ with xn 2 A for all n 2 N such that u ¼ limðxn Þ. 10. Establish the convergence or the divergence of the sequence ðyn Þ, where yn :¼ 1 1 1 þ þ þ nþ1 nþ2 2n for n 2 N: 11. Let xn :¼ 1=12 þ 1=22 þ þ 1=n2 for each n 2 N. Prove that ðxn Þ is increasing and bounded, and hence converges. [Hint: Note that if k 2, then 1=k2 1=kðk 1Þ ¼ 1=ðk 1Þ 1=k.] 12. Establish the convergence and find the limits of the following sequences. (a) ð1 þ 1=nÞnþ1 ; (b) ð1 þ 1=nÞ2n ; n 1 ; (d) ð1 1=nÞn : (c) 1þ nþ1 pffiffiffi 13. Use the method in Example 3.3.5 to calculate 2, correct to within 4 decimals. pffiffiffi 14. Use the method in Example 3.3.5 to calculate 5, correct to within 5 decimals. 15. Calculate the number en in Example 3.3.6 for n ¼ 2, 4, 8, 16. 16. Use a calculator to compute en for n ¼ 50; n ¼ 100, and n ¼ 1000. C03 12/08/2010 11:35:18 78 Page 78 CHAPTER 3 SEQUENCES AND SERIES Section 3.4 Subsequences and the Bolzano-Weierstrass Theorem In this section we will introduce the notion of a subsequence of a sequence of real numbers. Informally, a subsequence of a sequence is a selection of terms from the given sequence such that the selected terms form a new sequence. Usually the selection is made for a definite purpose. For example, subsequences are often useful in establishing the convergence or the divergence of the sequence. We will also prove the important existence theorem known as the Bolzano-Weierstrass Theorem, which will be used to establish a number of significant results. 3.4.1 Definition Let X ¼ ðxn Þ be a sequence of real numbers and let n1 < n2 < < nk < be a strictly increasing sequence of natural numbers. Then the sequence X 0 ¼ ðxnk Þ given by xn1 ; xn2 ; . . . ; xnk ; . . . is called a subsequence of X. For example, if X :¼ the subsequence 1 1 1 1;2;3;... X0 ¼ , then the selection of even indexed terms produces 1 1 1 1 ; ; ;...; ;... ; 2 4 6 2k where n1 ¼ 2; n2 ¼ 4; . . . ; nk ¼ 2k; : . . . Other subsequences of X ¼ ð1=nÞ are the following: 1 1 1 1 1 1 1 1 ; ; ;...; ;... ; ; ; ;...; ;... : 1 3 5 2k 1 2! 4! 6! ð2kÞ! The following sequences are not subsequences of X ¼ ð1=nÞ: 1 1 1 1 1 1 1 1 1 ; ; ; ; ; ;... ; ; 0; ; 0; ; 0; . . . : 2 1 4 3 6 5 1 3 5 A tail of a sequence (see 3.1.8) is a special type of subsequence. In fact, the m-tail corresponds to the sequence of indices n1 ¼ m þ 1; n2 ¼ m þ 2; . . . ; nk ¼ m þ k; . . . : But, clearly, not every subsequence of a given sequence need be a tail of the sequence. Subsequences of convergent sequences also converge to the same limit, as we now show. 3.4.2 Theorem If a sequence X ¼ ðxn Þ of real numbers converges to a real number x, then any subsequence X 0 ¼ ðxnk Þ of X also converges to x. Proof. Let e > 0 be given and let KðeÞ be such that if n KðeÞ, then jxn xj < e. Since n1 < n2 < < nk < is an increasing sequence of natural numbers, it is easily proved (by Induction) that nk k. Hence, if k KðeÞ, we also have nk k KðeÞ so that jxnk xj < e. Therefore the subsequence ðxnk Þ also converges to x. Q.E.D. 3.4.3 Examples (a) limðbn Þ ¼ 0 if 0 < b < 1. We have already seen, in Example 3.1.11(b), that if 0 < b < 1 and if xn :¼ bn , then it follows from Bernoulli’s Inequality that limðxn Þ ¼ 0. Alternatively, we see that since C03 12/08/2010 11:35:19 Page 79 3.4 SUBSEQUENCES AND THE BOLZANO-WEIERSTRASS THEOREM 79 0 < b < 1, then xnþ1 ¼ bnþ1 < bn ¼ xn so that the sequence ðxn Þ is decreasing. It is also clear that 0 xn 1, so it follows from the Monotone Convergence Theorem 3.3.2 that the sequence is convergent. Let x :¼ lim xn . Since ðx2n Þ is a subsequence of ðxn Þ it follows from Theorem 3.4.2 that x ¼ limðx2n Þ. Moreover, it follows from the relation x2n ¼ b2n ¼ ðbn Þ2 ¼ x2n and Theorem 3.2.3 that 2 x ¼ limðx2n Þ ¼ limðxn Þ ¼ x2 : Therefore we must have either x ¼ 0 or x ¼ 1. Since the sequence ðxn Þ is decreasing and bounded above by b < 1, we deduce that x ¼ 0. (b) limðc1=n Þ ¼ 1 for c > 1. This limit has been obtained in Example 3.1.11(c) for c > 0, using a rather ingenious argument. We give here an alternative approach for the case c > 1. Note that if zn :¼ c1=n ; then zn > 1 and znþ1 < zn for all n 2 N. (Why?) Thus by the Monotone Convergence Theorem, the limit z :¼ limðzn Þ exists. By Theorem 3.4.2, it follows that z ¼ limðz2n Þ. In addition, it follows from the relation z2n ¼ c1=2n ¼ ðc1=n Þ1=2 ¼ z1=2 n and Theorem 3.2.10 that z ¼ limðz2n Þ ¼ limðzn Þ 1=2 ¼ z1=2 : Therefore we have z2 ¼ z whence it follows that either z ¼ 0 or z ¼ 1. Since zn > 1 for all n 2 N, we deduce that z ¼ 1. & We leave it as an exercise to the reader to consider the case 0 < c < 1. The following result is based on a careful negation of the definition of limðxn Þ ¼ x. It leads to a convenient way to establish the divergence of a sequence. 3.4.4 Theorem Let X ¼ ðxn Þ be a sequence of real numbers, Then the following are equivalent: (i) The sequence X ¼ ðxn Þ does not converge to x 2 R. (ii) There exists an e0 > 0 such that for any k 2 N, there exists nk 2 N such that nk k and jxnk xj e0 . (iii) There exists an e0 > 0 and a subsequence X 0 ¼ ðxnk Þ of X such that jxnk xj e0 for all k 2 N. Proof. ðiÞ ) ðiiÞ If ðxn Þ does not converge to x, then for some e0 > 0 it is impossible to find a natural number k such that for all n k the terms xn satisfy jxn xj < e0. That is, for each k 2 N it is not true that for all n k the inequality jxn xj < e0 holds. In other words, for each k 2 N there exists a natural number nk k such that jxnk xj e0 . ðiiÞ ) ðiiiÞ Let e0 be as in (ii) and let n1 2 N be such that n1 1 and jxn1 xj e0 . Now let n2 2 N be such that n2 > n1 and jxn2 xj e0 ; let n3 2 N be such that n3 > n2 and jxn3 xj e0 . Continue in this way to obtain a subsequence X 0 ¼ ðxnk Þ of X such that jxnk xj e0 for all k 2 N. ðiiiÞ ) ðiÞ Suppose X ¼ ðxn Þ has a subsequence X 0 ¼ ðxnk Þ satisfying the condition in (iii). Then X cannot converge to x; for if it did, then, by Theorem 3.4.2, the subsequence X0 would also converge to x. But this is impossible, since none of the terms of X0 belongs to the e0-neighborhood of x. Q.E.D. C03 12/08/2010 11:35:19 80 Page 80 CHAPTER 3 SEQUENCES AND SERIES Since all subsequences of a convergent sequence must converge to the same limit, we have part (i) in the following result. Part (ii) follows from the fact that a convergent sequence is bounded. 3.4.5 Divergence Criteria If a sequence X ¼ ðxn Þ of real numbers has either of the following properties, then X is divergent. (i) X has two convergent subsequences X 0 ¼ ðxnk Þ and X 00 ¼ ðxrk Þ whose limits are not equal. (ii) X is unbounded. 3.4.6 Examples (a) The sequence X :¼ ðð1Þn Þ is divergent. The subsequence X 0 :¼ ðð1Þ2n Þ ¼ ð1; 1; . . .Þ converges to 1, and the subsequence 00 X :¼ ðð1Þ2n1 Þ ¼ ð1; 1; . . .Þ converges to 1. Therefore, we conclude from Theorem 3.4.5(i) that X is divergent. (b) The sequence 1; 12 ; 3; 14 ; . . . is divergent. This is the sequence Y ¼ ðyn Þ, where yn ¼ n if n is odd, and yn ¼ 1=n if n is even. It can easily be seen that Y is not bounded. Hence, by Theorem 3.4.5(ii), the sequence is divergent. (c) The sequence S :¼ (sin n) is divergent. This sequence is not so easy to handle. In discussing it we must, of course, make use of elementary properties of the sine function. We recall that sinðp=6Þ ¼ 12 ¼ sinð5p=6Þ and that sin x > 12 for x in the interval I 1 :¼ ðp=6; 5p=6Þ. Since the length of I 1 is 5p=6 p=6 ¼ 2p=3 > 2, there are at least two natural numbers lying inside I 1 ; we let n1 be the first such number. Similarly, for each k 2 N, sin x > 12 for x in the interval I k :¼ p=6 þ 2pðk 1Þ; 5p=6 þ 2pðk 1Þ : Since the length of Ik is greater than 2, there are at least two natural numbers lying inside Ik; we let nk be the first one. The subsequence S0 :¼ ðsin nk Þ of S obtained in this way has the property that all of its values lie in the interval 12 ; 1 . Similarly, if k 2 N and Jk is the interval J k :¼ 7p=6 þ 2pðk 1Þ; 11p=6 þ 2pðk 1Þ ; then it is seen that sin x < 12 for all x 2 J k and the length of J k is greater than 2. Let mk be the first natural number lying in Jk. Then the subsequence S 00 :¼ ðsin mk Þ of S has the property that all of its values lie in the interval 1; 12 . Given any real number c, it is readily seen that at least one of the subsequences S0 and 00 S lies entirely outside of the 12-neighborhood of c. Therefore c cannot be a limit of S. Since & c 2 R is arbitrary, we deduce that S is divergent. The Existence of Monotone Subsequences While not every sequence is a monotone sequence, we will now show that every sequence has a monotone subsequence. 3.4.7 Monotone Subsequence Theorem If X ¼ ðxn Þ is a sequence of real numbers, then there is a subsequence of X that is monotone. Proof. For the purpose of this proof, we will say that the mth term xm is a ‘‘peak’’ if xm xn for all n such that n m. (That is, xm is never exceeded by any term that follows it C03 12/08/2010 11:35:20 Page 81 3.4 SUBSEQUENCES AND THE BOLZANO-WEIERSTRASS THEOREM 81 in the sequence.) Note that, in a decreasing sequence, every term is a peak, while in an increasing sequence, no term is a peak. We will consider two cases, depending on whether X has infinitely many, or finitely many, peaks. Case 1: X has infinitely many peaks. In this case, we list the peaks by increasing subscripts: xm1 ; xm2 ; . . . ; xmk ; : . . . Since each term is a peak, we have xm1 xm2 xmk : Therefore, the subsequence ðxmk Þ of peaks is a decreasing subsequence of X. Case 2: X has a finite number (possibly zero) of peaks. Let these peaks be listed by increasing subscripts: xm1 ; xm2 ; . . . ; xmr . Let s1 :¼ mr þ 1 be the first index beyond the last peak. Since xs1 is not a peak, there exists s2 > s1 such that xs1 < xs2 . Since xs2 is not a peak, there exists s3 > s2 such that xs2 < xs3 . Continuing in this way, we obtain an increasing subsequence ðxsk Þ of X. Q.E.D. It is not difficult to see that a given sequence may have one subsequence that is increasing, and another subsequence that is decreasing. The Bolzano-Weierstrass Theorem We will now use the Monotone Subsequence Theorem to prove the Bolzano-Weierstrass Theorem, which states that every bounded sequence has a convergent subsequence. Because of the importance of this theorem we will also give a second proof of it based on the Nested Interval Property. 3.4.8 The Bolzano-Weierstrass Theorem A bounded sequence of real numbers has a convergent subsequence. First Proof. It follows from the Monotone Subsequence Theorem that if X ¼ ðxn Þ is a bounded sequence, then it has a subsequence X 0 ¼ ðxnk Þ that is monotone. Since this subsequence is also bounded, it follows from the Monotone Convergence Theorem 3.3.2 that the subsequence is convergent. Q.E.D. Second Proof. Since the set of values fxn : n 2 Ng is bounded, this set is contained in an interval I 1 :¼ ½a; b . We take n1 :¼ 1. We now bisect I1 into two equal subintervals I10 and I 100, and divide the set of indices fn 2 N : n > 1g into two parts: A1 :¼ fn 2 N : n > n1 ; xn 2 I10 g; B1 ¼ fn 2 N : n > n1 ; xn 2 I 100g: If A1 is infinite, we take I 2 :¼ I10 and let n2 be the smallest natural number in A1. If A1 is a finite set, then Bl must be infinite, and we take I 2 :¼ I 100 and let n2 be the smallest natural number in B1. We now bisect I2 into two equal subintervals I20 and I 200, and divide the set fn 2 N : n > n2 g into two parts: A2 ¼ fn 2 N : n > n2 ; xn ; 2 I20 g; B2 :¼ fn 2 N : n > n2 ; xn 2 I 200g If A2 is infinite, we take I 3 :¼ I20 and let n3 be the smallest natural number in A2. If A2 is a finite set, then B2 must be infinite, and we take I 3 :¼ I 200 and let n3 be the smallest natural number in B2. C03 12/08/2010 11:35:20 82 Page 82 CHAPTER 3 SEQUENCES AND SERIES We continue in this way to obtain a sequence of nested intervals I 1 I 2 I k and a subsequence ðxnk Þ of X such that xnk 2 I k for k 2 N. Since the length of I k is equal to ðb aÞ=2k1 , it follows from Theorem 2.5.3 that there is a (unique) common point j 2 I k for all k 2 N. Moreover, since xnk and j both belong to Ik, we have jxnk jj ðb aÞ=2k1 ; whence it follows that the subsequence ðxnk Þ of X converges to j. Q.E.D. Theorem 3.4.8 is sometimes called the Bolzano-Weierstrass Theorem for sequences, because there is another version of it that deals with bounded sets in R (see Exercise 11.2.6). It is readily seen that a bounded sequence can have various subsequences that converge to different limits or even diverge. For example, the sequence ðð1Þn Þ has subsequences that converge to 1, other subsequences that converge to þ1, and it has subsequences that diverge. Let X be a sequence of real numbers and let X0 be a subsequence of X. Then X0 is a sequence in its own right, and so it has subsequences. We note that if X 00 is a subsequence of X0 , then it is also a subsequence of X. 3.4.9 Theorem Let X ¼ ðxn Þ be a bounded sequence of real numbers and let x 2 R have the property that every convergent subsequence of X converges to x. Then the sequence X converges to x. Proof. Suppose M > 0 is a bound for the sequence X so that jxn j M for all n 2 N. If X does not converge to x, then Theorem 3.4.4 implies that there exist e0 > 0 and a subsequence X 0 ¼ ðxnk Þ of X such that ð1Þ jxnk xj e0 for all k 2 N: Since X 0 is a subsequence of X, the number M is also a bound for X 0. Hence the BolzanoWeierstrass Theorem implies that X 0 has a convergent subsequence X 00 . Since X 00 is also a subsequence of X, it converges to x by hypothesis. Thus, its terms ultimately belong to the e0-neighborhood of x, contradicting (1). Q.E.D. Limit Superior and Limit Inferior A bounded sequence of real numbers ðxn Þ may or may not converge, but we know from the Bolzano-Weierstrass Theorem 3.4.8 that there will be a convergent subsequence and possibly many convergent subsequences. A real number that is the limit of a subsequence of ðxn Þ is called a subsequential limit of ðxn Þ. We let S denote the set of all subsequential limits of the bounded sequence ðxn Þ. The set S is bounded, because the sequence is bounded. For example, if ðxn Þ is defined by xn :¼ ð1Þn þ 2=n, then the subsequence ðx2n Þ converges to 1, and the subsequence ðx2n1 Þ converges to 1. It is easily seen that the set of subsequential limits is S ¼ f1; 1g. Observe that the largest member of the sequence itself is x2 ¼ 2, which provides no information concerning the limiting behavior of the sequence. An extreme example is given by the set of all rational numbers in the interval [0, 1]. The set is denumerable (see Section 1.3) and therefore it can be written as a sequence ðrn Þ. Then it follows from the Density Theorem 2.4.8 that every number in [0, 1] is a subsequential limit of ðrn Þ. Thus we have S ¼ ½0; 1 . A bounded sequence ðxn Þ that diverges will display some form of oscillation. The activity is contained in decreasing intervals as follows. The interval ½t1 ; u1 , where t1 :¼ C03 12/08/2010 11:35:21 Page 83 3.4 SUBSEQUENCES AND THE BOLZANO-WEIERSTRASS THEOREM 83 inf fxn : n 2 Ng and u1 :¼ supfxn : n 2 Ng, contains the entire sequence. If for each m ¼ 1; 2; . . ., we define tm :¼ inffxn : n mg and um :¼ supfxn : n mg, the sequences ðtm Þ and ðum Þ are monotone and we obtain a nested sequence of intervals ½tm ; um where the mth interval contains the m-tail of the sequence. The preceding discussion suggests different ways of describing limiting behavior of a bounded sequence. Another is to observe that if a real number v has the property that xn > v for at most a finite number of values of n, then no subsequence of ðxn Þ can converge to a limit larger than v because that would require infinitely many terms of the sequence be larger than v. In other words, if v has the property that there exists Nv such that xn v for all n N v , then no number larger than v can be a subsequential limit of ðxn Þ. This observation leads to the following definition of limit superior. The accompanying definition of limit inferior is similar. 3.4.10 Definition Let X ¼ ðxn Þ be a bounded sequence of real numbers. (a) The limit superior of ðxn Þ is the infimum of the set V of v 2 R such that v < xn for at most a finite number of n 2 N. It is denoted by lim supðxn Þ or lim sup X or limðxn Þ: (b) The limit inferior of ðxn Þ is the supremum of the set of w 2 R such that xm < w for at most a finite number of m 2 N. It is denoted by lim infðxn Þ or lim inf X or limðxn Þ: For the concept of limit superior, we now show that the different approaches are equivalent. 3.4.11 Theorem If ðxn Þ is a bounded sequence of real numbers, then the following statements for a real number x are equivalent. (a) x ¼ lim supðxn Þ: (b) If e > 0, there are at most a finite number of n 2 N such that x þ e < xn , but an infinite number of n 2 N such that x e < xn . (c) If um ¼ supfxn : n mg; then x ¼ inffum : m 2 Ng ¼ limðum Þ. (d) If S is the set of subsequential limits of ðxn Þ, then x ¼ sup S. Proof. (a) implies (b). If e > 0, then the fact that x is an infimum implies that there exists a v in V such that x v < x þ e. Therefore x also belongs to V, so there can be at most a finite number of n 2 N such that x þ e < xn . On the other hand, x e is not in V so there are an infinite number of n 2 N such that x e < xn . (b) implies (c). If (b) holds, given e > 0, then for all sufficiently large m we have um < x þ e. Therefore, inffum : m 2 Ng x þ e. Also, since there are an infinite number of n 2 N such that x e < xn , then x e < um for all m 2 N and hence x e inffum : m 2 Ng. Since e > 0 is arbitrary, we conclude that x ¼ inffum : m 2 Ng. Moreover, since the sequence ðum Þ is monotone decreasing, we have infðum Þ ¼ limðum Þ. (c) implies (d). Suppose that X 0 ¼ ðxnk Þ is a convergent subsequence of X ¼ ðxn Þ: Since nk k, we have xnk uk and hence lim X 0 limðuk Þ ¼ x : Conversely, there exists n1 such that u1 1 xn1 u1 . Inductively choose nkþ1 > nk such that uk 1 < xnkþ1 uk : kþ1 Since lim ðuk Þ ¼ x , it follows that x ¼ limðxnk Þ, and hence x 2 S. C03 12/08/2010 11:35:22 84 Page 84 CHAPTER 3 SEQUENCES AND SERIES (d) implies (a). Let w ¼ sup S. If e > 0 is given, then there are at most finitely many n with w þ e < xn . Therefore w þ e belongs to V and lim sup ðxn Þ w þ e. On the other hand, there exists a subsequence of ðxn Þ converging to some number larger than w e, so that w e is not in V, and hence w e lim sup ðxn Þ. Since e > 0 is arbitrary, we conclude that w ¼ lim sup ðxn Þ. Q.E.D. As an instructive exercise, the reader should formulate the corresponding theorem for the limit inferior of a bounded sequence of real numbers. 3.4.12 Theorem A bounded sequence ðxn Þ is convergent if and only if lim sup ðxn Þ ¼ lim inf ðxn Þ. We leave the proof as an exercise. Other basic properties can also be found in the exercises. Exercises for Section 3.4 1. Give an example of an unbounded sequence that has a convergent subsequence. 2. Use the method of Example 3.4.3(b) to show that if 0 < c < 1, then limðc1=n Þ ¼ 1. 3. Let ð f n Þ be the Fibonacci sequence of Example 3.1.2(d), and let xn :¼ f nþ1 =f n . Given that limðxn Þ ¼ L exists, determine the value of L. 4. Show that the following sequences are divergent. (a) 1 ð1Þn þ 1=n ; (b) ðsin np=4Þ: 5. Let X ¼ ðxn Þ and Y ¼ ðyn Þ be given sequences, and let the ‘‘shuffled’’ sequence Z ¼ ðzn Þ be defined by z1 :¼ x1 ; z2 :¼ y1 ; . . . ; z2n1 :¼ xn ; z2n :¼ yn ; : . . . Show that Z is convergent if and only if both X and Y are convergent and lim X ¼ lim Y. 6. Let xn :¼ n1=n for n 2 N. (a) Show that xnþ1 < xn if and only if ð1 þ 1=nÞn < n, and infer that the inequality is valid for n 3. (See Example 3.3.6.) Conclude that ðxn Þ is ultimately decreasing and that x :¼ limðxn Þ exists. (b) Use the fact that the subsequence ðx2n Þ also converges to x to conclude that x ¼ 1. 7. Establish the convergence and find the limits of the following sequences: 2 (a) ð1 þ 1=n2 Þn ; (b) ð1 þ 1=2nÞn ; 2 (d) ð1 þ 2=nÞn : (c) ð1 þ 1=n2 Þ2n ; 8. Determine the limits of the following. (a) ð3nÞ1=2n ; (b) ð1 þ 1=2nÞ3n : 9. Suppose that every subsequence of X ¼ ðxn Þ has a subsequence that converges to 0. Show that lim X ¼ 0. 10. Let ðxn Þ be a bounded sequence and for each n 2 N let sn :¼ supfxk : k ng and S :¼ inffsn g. Show that there exists a subsequence of ðxn Þ that converges to S. 11. Suppose that xn 0 for all n 2 N and that lim ð1Þn xn exists. Show that ðxn Þ converges. 12. Show that if ðxn Þ is unbounded, then there exists a subsequence ðxnk Þ such that limð1=xnk Þ ¼ 0. 13. If xn :¼ ð1Þn =n, find the subsequence of ðxn Þ that is constructed in the second proof of the Bolzano-Weierstrass Theorem 3.4.8, when we take I 1 :¼ ½1; 1 . C03 12/08/2010 11:35:23 Page 85 3.5 THE CAUCHY CRITERION 85 14. Let ðxn Þ be a bounded sequence and let s :¼ supfxn : n 2 Ng. Show that if s 2 = fxn : n 2 Ng, then there is a subsequence of ðxn Þ that converges to s. 15. Let ðI n Þ be a nested sequence of closed bounded intervals. For each n 2 N, let xn 2 I n . Use the Bolzano-Weierstrass Theorem to give a proof of the Nested Intervals Property 2.5.2. 16. Give an example to show that Theorem 3.4.9 fails if the hypothesis that X is a bounded sequence is dropped. 17. Alternate the terms of the sequences (1 þ 1=n) and (1=n) to obtain the sequence ðxn Þ given by ð2; 1; 3=2; 1=2; 4=3; 1=3; 5=4; 1=4; . . .Þ: Determine the values of lim supðxn Þ and lim infðxn Þ. Also find supfxn g and inffxn g. 18. Show that if ðxn Þ is a bounded sequence, then ðxn Þ converges if and only if lim supðxn Þ ¼ lim infðxn Þ. 19. Show that if ðxn Þ and ðyn Þ are bounded sequences, then lim supðxn þ yn Þ lim supðxn Þ þ lim supðyn Þ: Give an example in which the two sides are not equal. Section 3.5 The Cauchy Criterion The Monotone Convergence Theorem is extraordinarily useful and important, but it has the significant drawback that it applies only to sequences that are monotone. It is important for us to have a condition implying the convergence of a sequence that does not require us to know the value of the limit in advance, and is not restricted to monotone sequences. The Cauchy Criterion, which will be established in this section, is such a condition. 3.5.1 Definition A sequence X ¼ ðxn Þ of real numbers is said to be a Cauchy sequence if for every e > 0 there exists a natural number HðeÞ such that for all natural numbers n; m HðeÞ, the terms xn ; xm satisfy jxn xm j < e. The significance of the concept of Cauchy sequence lies in the main theorem of this section, which asserts that a sequence of real numbers is convergent if and only if it is a Cauchy sequence. This will give us a method of proving a sequence converges without knowing the limit of the sequence. However, we will first highlight the definition of Cauchy sequence in the following examples. 3.5.2 Examples (a) The sequence (1=n) is a Cauchy sequence. If e > 0 is given, we choose a natural number H ¼ H(e) such that H > 2=e. Then if m; n H, we have 1=n 1=H < e=2 and similarly 1=m < e=2. Therefore, it follows that if m; n H, then 1 1 1 1 þ < e þ e ¼ e: n m n m 2 2 Since e > 0 is arbitrary, we conclude that (1=n) is a Cauchy sequence. (b) The sequence (1 þ (1)n) is not a Cauchy sequence. The negation of the definition of Cauchy sequence is: There exists e0 > 0 such that for every H there exist at least one n > H and at least one m > H such that jxn xm j e0 . For C03 12/08/2010 11:35:23 86 Page 86 CHAPTER 3 SEQUENCES AND SERIES the terms xn :¼ 1 þ ð1Þn , we observe that if n is even, then xn ¼ 2 and xnþ1 ¼ 0. If we take e0 ¼ 2, then for any H we can choose an even number n > H and let m :¼ n þ 1 to get jxn xnþ1 j ¼ 2 ¼ e0 : We conclude that (xn) is not a Cauchy sequence. & Remark We emphasize that to prove a sequence ðxn Þ is a Cauchy sequence, we may not assume a relationship between m and n, since the required inequality jxn xm j < e must hold for all n; m HðeÞ. But to prove a sequence is not a Cauchy sequence, we may specify a relation between n and m as long as arbitrarily large values of n and m can be chosen so that jxn xm j e0 . Our goal is to show that the Cauchy sequences are precisely the convergent sequences. We first prove that a convergent sequence is a Cauchy sequence. 3.5.3 Lemma If X ¼ ðxn Þ is a convergent sequence of real numbers, then X is a Cauchy sequence. Proof. If x :¼ lim X, then given e > 0 there is a natural number Kðe=2Þ such that if n Kðe=2Þ then jxn xj < e=2. Thus, if HðeÞ :¼ Kðe=2Þ and if n; m HðeÞ, then we have jxn xm j ¼ jðxn xÞ þ ðx xm Þj jxn xj þ jxm xj < e=2 þ e=2 ¼ e: Since e > 0 is arbitrary, it follows that (xn) is a Cauchy sequence. Q.E.D. In order to establish that a Cauchy sequence is convergent, we will need the following result. (See Theorem 3.2.2.) 3.5.4 Lemma A Cauchy sequence of real numbers is bounded. Proof. Let X :¼ ðxn Þ be a Cauchy sequence and let e :¼ 1. If H :¼ Hð1Þ and n H, then jxn xH j < 1. Hence, by the Triangle Inequality, we have jxn j jxH j þ 1 for all n H. If we set M :¼ supfjx1 j; jx2 j; . . . ; jxH1 j; jxH j þ 1g; then it follows that jxn j M for all n 2 N. Q.E.D. We now present the important Cauchy Convergence Criterion. 3.5.5 Cauchy Convergence Criterion A sequence of real numbers is convergent if and only if it is a Cauchy sequence. Proof. We have seen, in Lemma 3.5.3, that a convergent sequence is a Cauchy sequence. Conversely, let X ¼ ðxn Þ be a Cauchy sequence; we will show that X is convergent to some real number. First we observe from Lemma 3.5.4 that the sequence X is bounded. Therefore, by the Bolzano-Weierstrass Theorem 3.4.8, there is a subsequence X 0 ¼ ðxnk Þ of X that converges to some real number x. We shall complete the proof by showing that X converges to x. C03 12/08/2010 11:35:24 Page 87 3.5 THE CAUCHY CRITERION 87 Since X ¼ ðxn Þ is a Cauchy sequence, given e > 0 there is a natural number Hðe=2Þ such that if n; m Hðe=2Þ then jxn xm j < e=2: (1) Since the subsequence X 0 ¼ ðxnk Þ converges to x , there is a natural number K Hðe=2Þ belonging to the set fn1 ; n2 ; . . .g such that jxK x j < e=2: Since K Hðe=2Þ, it follows from (1) with m ¼ K that jxn xK j < e=2 n Hðe=2Þ: for Therefore, if n Hðe=2Þ, we have jxn x j ¼ jðxn xK Þ þ ðxK x Þj jxn xK j þ jxK x j < e=2 þ e=2 ¼ e: Sincee e > 0 is arbitrary, we infer that limðxn Þ ¼ x . Therefore the sequence X is convergent. Q.E.D. We will now give some examples of applications of the Cauchy Criterion. 3.5.6 Examples (a) Let X ¼ (xn) be defined by x1 :¼ 1; x2 :¼ 2; and 1 xn :¼ ðxn2 þ xn1 Þ for 2 n > 2: It can be shown by Induction that 1 xn 2 for all n 2 N. (Do so.) Some calculation shows that the sequence X is not monotone. However, since the terms are formed by averaging, it is readily seen that jxn xnþ1 j ¼ 1 2n1 for n 2 N: (Prove this by Induction.) Thus, if m > n, we may employ the Triangle Inequality to obtain jxn xm j jxn xnþ1 j þ jxnþ1 xnþ2 j þ þ jxm1 xm j ¼ ¼ 1 2 n1 1 2 n1 þ 1 1 n þ þ m2 2 2 1 1 1 þ þ þ mn1 2 2 < 1 2 n2 : Therefore, given e > 0, if n is chosen so large that 1=2n < e=4 and if m n, then it follows that jxn xm j < e. Therefore, X is a Cauchy sequence in R. By the Cauchy Criterion 3.5.5 we infer that the sequence X converges to a number x. To evaluate the limit x, we might first ‘‘pass to the limit’’ in the rule of definition xn ¼ 12 ðxn1 þ xn2 Þ to conclude that x must satisfy the relation x ¼ 12 ðx þ xÞ, which is true, but not informative. Hence we must try something else. C03 12/08/2010 11:35:24 88 Page 88 CHAPTER 3 SEQUENCES AND SERIES Since X converges to x, so does the subsequence X0 with odd indices. By Induction, the reader can establish that [see 1.2.4(f)] 1 1 1 x2nþ1 ¼ 1 þ þ 3 þ þ 2n1 2 2 2 2 1 1 n : ¼ 1þ 3 4 It follows from this (how?) that x ¼ lim X ¼ lim X 0 ¼ 1 þ 23 ¼ 53. (b) Let Y ¼ ðyn Þ be the sequence of real numbers given by 1 1 1 1 1 ð1Þnþ1 ; y2 :¼ ; . . . ; yn :¼ þ þ ; 1! 1! 2! 1! 2! n! Clearly, Y is not a monotone sequence. However, if m > n, then y1 :¼ ym yn ¼ ð1Þnþ2 ð1Þnþ3 ð1Þmþ1 þ þ þ : ðn þ 1Þ! ðn þ 2Þ! m! Since 2r1 r! [see 1.2.4(e)], it follows that if m > n, then (why?) jym yn j 1 1 1 þ þ þ ðn þ 1Þ! ðn þ 2Þ! m! 1 1 1 1 n þ nþ1 þ þ m1 < n1 : 2 2 2 2 Therefore, it follows that (yn) is a Cauchy sequence. Hence it converges to a limit y. At the present moment we cannot evaluate y directly; however, passing to the limit (with respect to m) in the above inequality, we obtain jyn yj 1=2n1 : Hence we can calculate y to any desired accuracy by calculating the terms yn for sufficiently large n. The reader should do this and show that y is approximately equal to 0.632 120 559. (The exact value of y is 1 1=e.) 1 1 1 þ þ þ (c) The sequence diverges. 1 2 n Let H :¼ ðhn Þ be the sequence defined by 1 1 1 þ þ þ for n 2 N; 1 2 n which was considered in 3.3.3(b). If m > n, then 1 1 þ þ : hm hn ¼ nþ1 m Since each of these m n terms exceeds 1=m, then hm hn > ðm nÞ=m ¼ 1 n=m. In particular, if m ¼ 2n we have h2n hn > 12. This shows that H is not a Cauchy sequence (why?); therefore H is not a convergent sequence. (In terms that will be introduced in Section 3.7, 1 X & 1=n is divergent.) we have just proved that the ‘‘harmonic series’’ hn :¼ n¼1 3.5.7 Definition We say that a sequence X ¼ ðxn Þ of real numbers is contractive if there exists a constant C; 0 < C < 1, such that jxnþ2 xnþ1 j Cjxnþ1 xn j for all n 2 N. The number C is called the constant of the contractive sequence. C03 12/08/2010 11:35:25 Page 89 3.5 THE CAUCHY CRITERION 89 3.5.8 Theorem Every contractive sequence is a Cauchy sequence, and therefore is convergent. Proof. If we successively apply the defining condition for a contractive sequence, we can work our way back to the beginning of the sequence as follows: jxnþ2 xnþ1 j Cjxnþ1 xn j C2 jxn xn1 j C3 jxn1 xn2 j Cn jx2 x1 j: For m > n, we estimate jxm xn j by first applying the Triangle Inequality and then using the formula for the sum of a geometric progression (see 1.2.4(f)). This gives jxm xn j jxm xm1 j þ jxm1 xm2 j þ þ jxnþ1 xn j Cm2 þ C m3 þ þ Cn1 jx2 x1 j 1 Cmn ¼ Cn1 jx2 x1 j 1C 1 n1 C jx2 x1 j: 1C Since 0 < C < 1, we know limðC n Þ ¼ 0 [see 3.1.11(b)]. Therefore, we infer that (xn) is a Cauchy sequence. It now follows from the Cauchy Convergence Criterion 3.5.5 that (xn) is Q.E.D. a convergent sequence. 3.5.9 Example We consider the sequence of Fibonacci fractions xn :¼ f n =f nþ1 where f 1 ¼ f 2 ¼ 1 and f nþ1 ¼ f n þ f n1 . (See Example 3.1.2(d).) The first few terms are x1 ¼ 1; x2 ¼ 1=2; x3 ¼ 2=3; x4 ¼ 3=5; x5 ¼ 5=8, and so on. It is shown that the sequence ðxn Þ is given inductively by the equation xnþ1 ¼ 1=ð1 þ xn Þ as follows: xnþ1 ¼ f nþ1 f nþ1 1 1 ¼ ¼ ¼ : 1 þ xn f nþ2 f nþ1 þ f n 1 þ f n f nþ1 An induction argument establishes 1=2 xn 1 for all n, so that adding 1 and taking reciprocals gives us the inequality 1=2 1=ð1 þ xn Þ 2=3 for all n. It then follows that jxnþ1 xn j ¼ jxn xn1 j 2 2 4 jxn xn1 j ¼ jxn xn1 j: ð1 þ xn Þð1 þ xn1 Þ 3 3 9 Hence, the sequence (xn) is contractive and therefore converges by Theorem 3.5.8. Passing to the limit x ¼ limðxn Þ, we obtain the equation x ¼ 1=ð1 þ xÞ, so that x satisfies the 2 equation xp þ ffiffiffi x 1 ¼ 0. The quadratic formula gives us the positive solution x ¼ ð1 þ 5Þ=2 ¼ 0:618034:p. ffiffi.ffi. The reciprocal 1=x ¼ ð1 þ 5Þ=2 ¼ 1:618034 . . . is often denoted by the Greek letter w and referred to as the Golden Ratio in the history of geometry. In the artistic theory of the ancient Greek philosophers, a rectangle having w as the ratio of the longer side to the shorter side is the rectangle most pleasing to the eye. The number also has many interesting mathematical & properties. (A historical discussion of the Golden Ratio can be found on Wikipedia.) In the process of calculating the limit of a contractive sequence, it is often very important to have an estimate of the error at the nth stage. In the next result we give two C03 12/08/2010 11:35:26 90 Page 90 CHAPTER 3 SEQUENCES AND SERIES such estimates: the first one involves the first two terms in the sequence and n; the second one involves the difference xn xn1 . 3.5.10 Corollary If X :¼ ðxn Þ is a contractive sequence with constant C; 0 < C < 1, and if x :¼ lim X, then n1 (i) jx xn j C jx x1 j; 1C 2 (ii) jx xn j C jxn xn1 j: 1C Proof. From the preceding proof, if m > n, then jxm xn j Cn1 =ð1 CÞ jx2 x1 j. If we let m ! 1 in this inequality, we obtain (i). To prove (ii), recall that if m > n, then jxm xn j jxm xm1 j þ þ jxnþ1 xn j: Since it is readily established, using Induction, that jxnþk xnþk1 j C k jxn xn1 j; we infer that jxm xn j ðC mn þ þ C2 þ CÞjxn xn1 j C jxn xn1 j 1C We now let m ! 1 in this inequality to obtain assertion (ii). Q.E.D. 3.5.11 Example We are told that the cubic equation x3 7x þ 2 ¼ 0 has a solution between 0 and 1 and we wish to approximate this solution. This can be accomplished by means of an iteration procedure as follows. We first rewrite the equation as x ¼ ðx3 þ 2Þ=7 and use this to define a sequence. We assign to x1 an arbitrary value between 0 and 1, and then define xnþ1 :¼ 17 ðx3n þ 2Þ for n 2 N: Because 0 < x1 < 1, it follows that 0 < xn < 1 for all n 2 N. (Why?) Moreover, we have 1 1 1 xnþ2 xnþ1 ¼ 7ðx3nþ1 þ 2Þ 7ðx3n þ 2Þ ¼ 7x3nþ1 x3n ¼ 17x2nþ1 þ xnþ1 xn þ x2n xnþ1 xn 37xnþ1 xn : Therefore, (xn) is a contractive sequence and hence there exists r such that limðxn Þ ¼ r. If we pass to the limit on both sides of the equality xnþ1 ¼ ðx3n þ 2Þ=7, we obtain r ¼ ðr3 þ 2Þ=7 and hence r3 7r þ 2 ¼ 0. Thus r is a solution of the equation. We can approximate r by choosing x1, and calculating x2 ; x3 ; . . . successively. For example, if we take x1 ¼ 0:5, we obtain (to nine decimal places): x2 ¼ 0:303 571 429; x3 ¼ 0:289 710 830; x4 ¼ 0:289 188 016; x5 ¼ 0:289 169 244; x6 ¼ 0:289 168 571; etc: C03 12/08/2010 11:35:27 Page 91 3.6 PROPERLY DIVERGENT SEQUENCES 91 To estimate the accuracy, we note that jx2 x1 j < 0:2. Thus, after n steps it follows from Corollary 3.5.10(i) that we are sure that jx xn j 3n1 ð7n2 20Þ. Thus, when n ¼ 6, we are sure that jx x6 j 35 =ð74 20Þ ¼ 243=48 020 < 0:0051: Actually the approximation is substantially better than this. In fact, since jx6 x5 j < 0:000 0005, it follows from 3.5.10(ii) that jx x6 j 34 jx6 x5 j < 0:000 0004. Hence the & first five decimal places of x6 are correct. Exercises for Section 3.5 1. Give an example of a bounded sequence that is not a Cauchy sequence. 2. Show directly from the definition that the following are Cauchy sequences. nþ1 1 1 (a) ; (b) 1 þ þ þ . n 2! n! 3. Show directly from the definition that the following are not Cauchy sequences. ð1Þn ; (c) (ln n) (a) ð1Þn ; (b) nþ n 4. Show directly from the definition that if (xn) and (yn) are Cauchy sequences, then (xn þ yn) and ðxn yn Þ are Cauchy sequences. pffiffiffi 5. If xn :¼ n, show that (xn) satisfies limjxnþ1 xn j ¼ 0, but that it is not a Cauchy sequence. 6. Let p be a given natural number. Give an example of a sequence (xn) that is not a Cauchy sequence, but that satisfies limjxnþp xn j ¼ 0. 7. Let (xn) be a Cauchy sequence such that xn is an integer for every n 2 N. Show that (xn) is ultimately constant. 8. Show directly that a bounded, monotone increasing sequence is a Cauchy sequence. 9. If 0 < r < 1 and jxnþ1 xn j < rn for all n 2 N, show that (xn) is a Cauchy sequence. 10. If x1 < x2 are arbitrary real numbers and xn :¼ 12 ðxn2 þ xn1 Þ for n > 2, show that (xn) is convergent. What is its limit? 11. If y1 < y2 are arbitrary real numbers and yn :¼ 13 yn1 þ 23 yn2 for n > 2, show that (yn) is convergent. What is its limit? 12. If x1 > 0 and xnþ1 :¼ ð2 þ xn Þ1 for n 1, show that (xn) is a contractive sequence. Find the limit. 13. If x1 :¼ 2 and xnþ1 :¼ 2 þ 1=xn for n 1, show that (xn) is a contractive sequence. What is its limit? 14. The polynomial equation x3 5x þ 1 ¼ 0 has a root r with 0 < r < 1. Use an appropriate contractive sequence to calculate r within 104. Section 3.6 Properly Divergent Sequences For certain purposes it is convenient to define what is meant for a sequence ðxn Þ of real numbers to ‘‘tend to 1.’’ C03 12/08/2010 11:35:28 92 Page 92 CHAPTER 3 SEQUENCES AND SERIES 3.6.1 Definition Let ðxn Þ be a sequence of real numbers. (i) We say that ðxn Þ tends to 1, and write limðxn Þ ¼ þ1, if for every a 2 R there exists a natural number KðaÞ such that if n KðaÞ, then xn > a. (ii) We say that (xn) tends to 1, and write limðxn Þ ¼ 1, if for every b 2 R there exists a natural number K(b) such that if n KðbÞ, then xn < b. We say that ðxn Þ is properly divergent in case we have either limðxn Þ ¼ þ1 or limðxn Þ ¼ 1. The reader should realize that we are using the symbols þ1 and 1 purely as a convenient notation in the above expressions. Results that have been proved in earlier sections for conventional limits limðxn Þ ¼ L ðfor L 2 RÞ may not remain true when limðxn Þ ¼ 1. 3.6.2 Examples (a) limðnÞ ¼ þ1. In fact, if a 2 R is given, let K(a) be any natural number such that K(a) > a. (b) limðn2 Þ ¼ þ1. If K(a) is a natural number such that K(a) > a, and if n KðaÞ then we have n2 n > a. (c) If c > 1, then limðcn Þ ¼ þ1. Let c ¼ 1 þ b, where b > 0. If a 2 R is given, let K(a) be a natural number such that KðaÞ > a=b. If n KðaÞ it follows from Bernoulli’s Inequality that cn ¼ ð1 þ bÞn 1 þ nb > 1 þ a > a: & Therefore limðcn Þ ¼ þ1. Monotone sequences are particularly simple in regard to their convergence. We have seen in the Monotone Convergence Theorem 3.3.2 that a monotone sequence is convergent if and only if it is bounded. The next result is a reformulation of that result. 3.6.3 Theorem A monotone sequence of real numbers is properly divergent if and only if it is unbounded. (a) If ðxn Þ is an unbounded increasing sequence, then limðxn Þ ¼ þ1. (b) If ðxn Þ is an unbounded decreasing sequence, then limðxn Þ ¼ 1. Proof. (a) Suppose that (xn) is an increasing sequence. We know that if (xn) is bounded, then it is convergent. If (xn) is unbounded, then for any a 2 R there exists nðaÞ 2 N such that a < xnðaÞ . But since (xn) is increasing, we have a < xn for all n nðaÞ. Since a is arbitrary, it follows that limðxn Þ ¼ þ1. Part (b) is proved in a similar fashion. Q.E.D. The following ‘‘comparison theorem’’ is frequently used in showing that a sequence is properly divergent. [In fact, we implicitly used it in Example 3.6.2(c).] 3.6.4 Theorem Let (xn) and (yn) be two sequences of real numbers and suppose that (1) xn yn for all (a) If limðxn Þ ¼ þ1, then limðyn Þ ¼ þ1. (b) If limðyn Þ ¼ 1, then limðxn Þ ¼ 1. n 2 N: C03 12/08/2010 11:35:30 Page 93 3.6 PROPERLY DIVERGENT SEQUENCES 93 Proof. (a) If limðxn Þ ¼ þ1, and if a 2 R is given, then there exists a natural number K(a) such that if n KðaÞ, then a < xn . In view of (1), it follows that a < yn for all n KðaÞ. Since a is arbitrary, it follows that limðyn Þ ¼ þ1. The proof of (b) is similar. Q.E.D. Remarks (a) Theorem 3.6.4 remains true if condition (1) is ultimately true; that is, if there exists m 2 N such that xn yn for all n m. (b) If condition (1) of Theorem 3.6.4 holds and if limðyn Þ ¼ þ1, it does not follow that limðxn Þ ¼ þ1. Similarly, if (1) holds and if limðxn Þ ¼ 1, it does not follow that limðyn Þ ¼ 1. In using Theorem 3.6.4 to show that a sequence tends to þ1 [respectively, 1] we need to show that the terms of the sequence are ultimately greater [respectively, less] than or equal to the corresponding terms of a sequence that is known to tend to þ1 [respectively, 1]. Since it is sometimes difficult to establish an inequality such as (1), the following ‘‘limit comparison theorem’’ is often more convenient to use than Theorem 3.6.4. 3.6.5 Theorem Let (xn) and (yn) be two sequences of positive real numbers and suppose that for some L 2 R; L > 0, we have limðxn =yn Þ ¼ L: (2) Then limðxn Þ ¼ þ1 if and only if limðyn Þ ¼ þ1. Proof. If (2) holds, there exists K 2 N such that 1 2L < xn =yn < 32 L for all n K: Hence we have 12 L yn < xn < 32 L yn for all n K. The conclusion now follows from a slight modification of Theorem 3.6.4. We leave the details to the reader. Q.E.D. The reader can show that the conclusion need not hold if either L ¼ 0 or L ¼ þ1. However, there are some partial results that can be established in these cases, as will be seen in the exercises. Exercises for Section 3.6 1. Show that if (xn) is an unbounded sequence, then there exists a properly divergent subsequence. 2. Give examples of properly divergent sequences (xn) and (yn) with yn 6¼ 0 for all n 2 N such that: (a) ðxn =yn Þ is convergent, (b) ðxn =yn Þ is properly divergent. 3. Show that if xn > 0 for all n 2 N, then limðxn Þ ¼ 0 if and only if limð1=xn Þ ¼ þ1. 4. Establish pffiffiffiffiffiffiffiffiffiffiffi pffiffiffi the proper divergence of the following sequences. (a) ð nÞ; (b) ð n þ 1Þ; pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi (c) ð n 1Þ; (d) ðn= n þ 1Þ: 5. Is the sequence (n sin n) properly divergent? 6. Let (xn) be properly divergent and let (yn) be such that limðxn yn Þ belongs to R. Show that (yn) converges to 0. 7. Let (xn) and (yn) be sequences of positive numbers such that limðxn =yn Þ ¼ 0. (a) Show that if limðxn Þ ¼ þ1, then limðyn Þ ¼ þ1. (b) Show that if (yn) is bounded, then limðxn Þ ¼ 0. C03 12/08/2010 11:35:31 94 Page 94 CHAPTER 3 SEQUENCES AND SERIES 8. Investigate the convergence or the divergence of the following sequences: pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi (a) ð n2 þ 2Þ; (b) ð n=ðn2 þ 1ÞÞ; pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi (c) ð n2 þ 1= nÞ; (d) ðsin nÞ: 9. Let ðxn Þ and ðyn Þ be sequences of positive numbers such that limðxn =yn Þ ¼ þ1, (a) Show that if limðyn Þ ¼ þ1, then limðxn Þ ¼ þ1. (b) Show that if (xn) is bounded, then limðyn Þ ¼ 0. 10. Show that if limðan =nÞ ¼ L, where L > 0, then limðan Þ ¼ þ1. Section 3.7 Introduction to Infinite Series We will now give a brief introduction to infinite series of real numbers. This is a topic that will be discussed in more detail in Chapter 9, but because of its importance, we will establish a few results here. These results will be seen to be immediate consequences of theorems we have met in this chapter. In elementary texts, an infinite series is sometimes ‘‘defined’’ to be ‘‘an expression of the form’’ (1) x 1 þ x 2 þ þ xn þ : However, this ‘‘definition’’ lacks clarity, since there is a priori no particular value that we can attach to this array of symbols, which calls for an infinite number of additions to be performed. 3.7.1 Definition If X :¼ (xn) is a sequence in R, then the infinite series (or simply the series) generated by X is the sequence S :¼ (sk) defined by s1 :¼ x1 s2 :¼ s1 þ x2 ð¼ x1 þ x2 Þ sk :¼ sk1 þ xk ð¼ x1 þ x2 þ þ xk Þ The numbers xn are called the terms of the series and the numbers sk are called the partial sums of this series. If lim S exists, we say that this series is convergent and call this limit the sum or the value of this series. If this limit does not exist, we say that the series S is divergent. It is convenient to use symbols such as X X ðxn Þ or xn (2) or 1 X xn n¼1 to denote both the infinite series S generated by the sequence X ¼ (xn) and also to denote the value lim S, in case this limit exists. Thus the symbols in (2) may be regarded merely as a way of exhibiting an infinite series whose convergence or divergence is to be investigated. In practice, this double use of these notations does not lead to any confusion, provided it is understood that the convergence (or divergence) of the series must be established. C03 12/08/2010 11:35:31 Page 95 3.7 INTRODUCTION TO INFINITE SERIES 95 Just as a sequence may be indexed such that its first element is not x1, but is x0, or x5 or x99, we will denote the series having these numbers as their first element by the symbols 1 X xn or n¼0 1 X xn or n¼5 1 X xn : n¼99 It should be noted that when the first term in the series is xN, then the first partial sum is denoted by sN. Warning The reader should guard against confusing the words ‘‘sequence’’ and ‘‘series.’’ In nonmathematical language, these words are interchangeable; however, in mathematics, these words are not synonyms. Indeed, a series is a sequence S ¼ (sk) obtained from a given sequence X ¼ (xn) according to the special procedure given in Definition 3.7.1. 3.7.2 Examples (a) Consider the sequence X :¼ ðrn Þ1 n¼0 where r 2 R, which generates the geometric series: 1 X (3) rn ¼ 1 þ r þ r2 þ þ rn þ : n¼0 We will show that if jrj < 1, then this series converges to 1=ð1 rÞ. (See also Example 1.2.4(f).) Indeed, if sn :¼ 1 þ r þ r2 þ þ rn for n 0, and if we multiply sn by r and subtract the result from sn, we obtain (after some simplification): sn ð1 rÞ ¼ 1 rnþ1 : Therefore, we have sn 1 rnþ1 ¼ ; 1r 1r from which it follows that nþ1 sn 1 jrj : 1r j1 rj Since jrjnþ1 ! 0 when jrj < 1, it follows that the geometric series converges to 1=ð1 rÞ when jrj < 1. 1 (b) Consider the series generated by ð1Þn n¼0 ; that is, the series: (4) 1 X ð1Þn ¼ ðþ1Þ þ ð1Þ þ ðþ1Þ þ ð1Þ þ : n¼0 It is easily seen (by Mathematical Induction) that sn ¼ 1 if n 0 is even and sn ¼ 0 if n is odd; therefore, the sequence of partial sums is ð1; 0; 1; 0; . . .Þ. Since this sequence is not convergent, the series (4) is divergent. (c) Consider the series (5) 1 X 1 1 1 1 ¼ þ þ þ : nðn þ 1Þ 1 2 2 3 3 4 n¼1 C03 12/08/2010 11:35:32 96 Page 96 CHAPTER 3 SEQUENCES AND SERIES By a stroke of insight, we note that 1 1 1 ¼ : kðk þ 1Þ k k þ 1 Hence, on adding these terms from k ¼ 1 to k ¼ n and noting the telescoping that takes place, we obtain 1 1 sn ¼ ; 1 nþ1 whence it follows that sn ! 1. Therefore the series (5) converges to 1. & We now present a very useful and simple necessary condition for the convergence of a series. It is far from being sufficient, however. 3.7.3 The nth Term Test If the series P xn converges, then limðxn Þ ¼ 0. P Proof. By Definition 3.7.1, the convergence of xn requires that limðsk Þ exists. Since xn ¼ sn sn1 , then limðxn Þ ¼ limðsn Þ limðsn1 Þ ¼ 0. Q.E.D. Since the following Cauchy Criterion is precisely a reformulation of Theorem 3.5.5, we will omit its proof. P 3.7.4 Cauchy Criterion for Series The series xn converges if and only if for every e > 0 there exists MðeÞ 2 N such that if m > n MðeÞ, then (6) jsm sn j ¼ jxnþ1 þ xnþ2 þ þ xm j < e: The next result, although limited in scope, is of great importance and utility. P 3.7.5 Theorem Let (xn) be a sequence of nonnegative real numbers. Then the series xn converges if and only if the sequence S ¼ ðsk Þ of partial sums is bounded. In this case, 1 X xn ¼ limðsk Þ ¼ supfsk : k 2 Ng: n¼1 Proof. Since xn 0, the sequence S of partial sums is monotone increasing: s1 s2 sk : By the Monotone Convergence Theorem 3.3.2, the sequence S ¼ ðsk Þ converges if and only if it is bounded, in which case its limit equals supfsk g. Q.E.D. 3.7.6 Examples (a) The geometric series (3) diverges if jrj 1. This follows from the fact that the terms rn do not approach 0 when jrj 1. 1 X 1 diverges. (b) The harmonic series n n¼1 Since the terms 1=n ! 0, we cannot use the nth Term Test 3.7.3 to establish this divergence. However, it was seen in Examples 3.3.3(b) and 3.5.6(c) that the sequence (sn) of partial sums is not bounded. Therefore, it follows from Theorem 3.7.5 that the harmonic series is divergent. This series is famous for the very slow growth of its partial sums (see the C03 12/08/2010 11:35:33 Page 97 3.7 INTRODUCTION TO INFINITE SERIES 97 discussion in Example 3.3.3(b)) and also for the variety of proofs of its divergence. Here is a proof by contradiction. If we assume the series converges to the number S, then we have 1 1 1 1 1 1 1 þ þ þ S ¼ 1þ þ þ þ þ þ 2 3 4 5 6 2n 1 2n 1 1 1 1 1 1 1 1 þ þ þ þ > þ þ þ þ þ 2 2 4 4 6 6 2n 2n 1 1 1 ¼ 1 þ þ þ þ þ 2 3 n ¼ S: The contradiction S > S shows the assumption of convergence must be false and the harmonic series must diverge. Note The harmonic series receives its musical name from the fact that the wavelengths of the overtones of a vibrating string are 1=2, 1=3, 1=4, . . . , of the string’s fundamental wavelength. 1 X 1 (c) The 2-series is convergent. n2 n¼1 Since the partial sums are monotone, it suffices (why?) to show that some subsequence of (sk) is bounded. If k1 :¼ 21 1 ¼ 1, then sk1 ¼ 1. If k2 :¼ 22 1 ¼ 3, then 1 1 1 2 1 sk 2 ¼ þ 2 þ 2 < 1 þ 2 ¼ 1 þ ; 1 2 2 3 2 and if k3 :¼ 23 1 ¼ 7, then we have 1 1 1 1 4 1 1 sk3 ¼ sk2 þ 2 þ 2 þ 2 þ 2 < sk2 þ 2 < 1 þ þ 2 : 2 2 7 4 4 6 5 By Mathematical Induction, we find that if kj :¼ 2j 1, then 2 j1 0 < skj < 1 þ 12 þ 12 þ þ 12 : Since the term on the right is a partial sum of a geometric series with r ¼ 12, it is dominated by 1= 1 12 ¼ 2, and Theorem 3.7.5 implies that the 2-series converges. 1 X 1 (d) The p-series converges when p > 1. np n¼1 Since the argument is very similar to the special case considered in part (c), we will leave some of the details to the reader. As before, if k1 :¼ 21 1 ¼ 1, then sk1 ¼ 1. If k2 :¼ 22 1 ¼ 3, then since 2p < 3p , we have 1 1 1 2 1 sk2 ¼ p þ p þ p < 1 þ p ¼ 1 þ p1 : 1 2 3 2 2 Further, if k3 :¼ 23 1, then (how?) it is seen that 4 1 1 p < 1 þ p1 þ p1 : 4 2 4 Finally, we let r :¼ 1=2p1 ; since p > 1, we have 0 < r < 1. Using Mathematical Induction, we show that if kj :¼ 2j 1, then 1 : 0 < skj < 1 þ r þ r2 þ þ r j1 < 1r Therefore, Theorem 3.7.5 implies that the p-series converges when p > 1. sk3 < sk2 þ C03 12/08/2010 11:35:34 98 Page 98 CHAPTER 3 SEQUENCES AND SERIES (e) The p-series 1 X 1 diverges when 0 < p 1. p n n¼1 We will use the elementary inequality np n when n 2 N and 0 < p 1. It follows that 1 1 p for n 2 N: n n Since the partial sums of the harmonic series are not bounded, this inequality shows that the partial sums of the p-series are not bounded when 0 < p 1. Hence the p-series diverges for these values of p. (f) The alternating harmonic series, given by (7) 1 X ð1Þnþ1 n¼1 n ¼ 1 1 1 ð1Þnþ1 þ þ þ 1 2 3 n is convergent. The reader should compare this series with the harmonic series in (b), which is divergent. Thus, the subtraction of some of the terms in (7) is essential if this series is to converge. Since we have 1 1 1 1 1 1 s2n ¼ þ þ þ ; 1 2 3 4 2n 1 2n it is clear that the ‘‘even’’ subsequence ðs2n Þ is increasing. Similarly, the ‘‘odd’’ subsequence ðs2nþ1 Þ is decreasing since 1 1 1 1 1 1 1 : s2nþ1 ¼ 1 2 3 4 5 2n 2n þ 1 Since 0 < s2n < s2n þ 1=ð2n þ 1Þ ¼ s2nþ1 1, both of these subsequences are bounded below by 0 and above by 1. Therefore they are both convergent and to the same value. Thus the sequence (sn) of partial sums converges, proving that the alternating harmonic series (7) & converges. (It is far from obvious that the limit of this series is equal to ln 2.) Comparison Tests Our first test shows that if the terms of a nonnegative series are dominated by the corresponding terms of a convergent series, then the first series is convergent. 3.7.7 Comparison Test Let X :¼ (xn) and Y :¼ (yn) be real sequences and suppose that for some K 2 N we have (8) 0 xn yn for n K: P P (a) Then the convergence of yn implies the convergence of xn . P P (b) The divergence of xn implies the divergence of yn . Proof. (a) Suppose that m > n MðeÞ, then P yn converges and, given e > 0, let MðeÞ 2 N be such that if ynþ1 þ þ ym < e: C03 12/08/2010 11:35:34 Page 99 3.7 INTRODUCTION TO INFINITE SERIES 99 If m > supfK; MðeÞg, then it follows that 0 xnþ1 þ þ xm ynþ1 þ þ ym < e; P from which the convergence of xn follows. (b) This statement is the contrapositive of (a). Q.E.D. Since it is sometimes difficult to establish the inequalities (8), the next result is frequently very useful. 3.7.8 Limit Comparison Test Suppose that X :¼ ðxn Þ and Y :¼ ðyn Þ are strictly positive sequences and suppose that the following limit exists in R: xn (9) r :¼ lim : yn P P (a) If r 6¼ 0 then xn is convergent if and only if yn is convergent. P P (b) If r ¼ 0 and if yn is convergent, then xn is convergent. Proof. (a) It follows from (9) and Exercise 3.1.18 that there exists K 2 N such that 12 r xn =yn 2r for n K, whence 1 2 r yn xn ð2rÞyn for n K: If we apply the Comparison Test 3.7.7 twice, we obtain the assertion in (a). (b) If r ¼ 0, then there exists K 2 N such that 0 < xn yn for n K; so that Theorem 3.7.7(a) applies. Q.E.D. Remark The Comparison Tests 3.7.7 and 3.7.8 depend on having a stock of series that one knows to be convergent (or divergent). The reader will find that the p-series is often useful for this purpose. 3.7.9 Examples (a) The series 1 X n¼1 n2 1 converges. þn It is clear that the inequality 0< n2 1 1 < 2 þn n for n2N P is valid. Since the series 1=n2 is convergent (by Example 3.7.6(c)), we can apply the Comparison Test 3.7.7 to obtain the convergence of the given series. 1 X 1 is convergent. (b) The series 2nþ1 n n¼1 If the inequality (10) n2 1 1 2 nþ1 n C03 12/08/2010 11:35:35 100 Page 100 CHAPTER 3 SEQUENCES AND SERIES were true, we could argue as in (a). However, (10) is false for all n 2 N. The reader can probably show that the inequality 1 2 2 0< 2 n nþ1 n is valid for all n 2 N, and this inequality will work just as well. However, it might take some experimentation to think of such an inequality and then establish it. Instead, if we take xn :¼ 1=ðn2 n þ 1Þ and yn :¼ 1=n2 ; then we have xn n2 1 ¼ ! 1: ¼ 2 yn n n þ 1 1 ð1=nÞ þ ð1=n2 Þ Therefore, the convergence of the given series follows from the Limit Comparison Test 3.7.8(a). 1 X 1 pffiffiffiffiffiffiffiffiffiffiffi is divergent. (c) The series nþ1 n¼1 P pffiffiffi This series closely resembles the series 1= np , ffiffiffiffiffiffiffiffiffiffiffi which is a p-seriespffiffiwith p ¼ 12; by ffi Example 3.7.6(e), it is divergent. If we let xn :¼ 1= n þ 1 and yn :¼ 1= n, then we have pffiffiffi n xn 1 ¼ pffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! 1: yn nþ1 1 þ 1=n Therefore the Limit Comparison Test 3.7.8(a) applies. 1 X 1 is convergent. (d) The series n! n¼1 It would be possible to establish this convergence by showing (by Induction) that n2 < n! for n 4, whence it follows that 1 1 < 2 for n 4: n! n Alternatively, if we let x :¼ 1=n! and yn :¼ 1=n2 , then (when n 4) we have 0< 0 x n n2 n 1 < ! 0: ¼ ¼ yn n! 1 2 ðn 1Þ n 2 Therefore the Limit Comparison Test 3.7.8(b) applies. (Note that this test was a bit troublesome to apply since we do not presently know the convergence of any series for which the limit of xn =yn is really easy to determine.) & Exercises for Section 3.7 P P 1. Let an be a given P series and let bn be the series in which the terms are the same and in the same order as in an except that the P P terms for which an ¼ 0 have been omitted. Show that an converges to A if and only if bn converges to A. 2. Show that the convergence of a series is not affected by changing a finite number of its terms. (Of course, the value of the sum may be changed.) 3. By using partial fractions, show that 1 1 X X 1 1 1 ¼ 1; (b) ¼ > 0; if a > 0: (a) ð n þ 1 Þ ð n þ 2 Þ ð a þ n Þ ð a þ n þ 1 Þ a n¼0 n¼0 1 X 1 1 (c) ¼ : nðn þ 1Þðn þ 2Þ 4 n¼1 P P P yn are convergent, show that ðxn þ yn Þ is convergent. 4. If xn and C03 12/08/2010 11:35:36 Page 101 3.7 INTRODUCTION TO INFINITE SERIES 101 P P 5. P Can you give an example of a convergent series xn and a divergent series yn such that ðxn þ yn Þ is convergent? Explain. 1 P ð2=7Þn . (Note the series starts at n ¼ 2.) 6. (a) Calculate the value of (b) Calculate the value of n¼2 1 P ð1=3Þ2n . (Note the series starts at n ¼ 1.) n¼1 1 P 7. Find a formula for the series n¼1 r2n when jrj < 1. 8. Let r1 ; r2 ; . . . ; rn ; . . . be an enumeration of the rational numbers in the interval [0,1]. (See Section 1.3.) For a given e > 0, put an interval of length en about the nth rational number rn for n ¼ 1; 2; 3; . . . ; and find the total sum of the lengths of all the intervals. Evaluate this number for e ¼ 0:1 and e ¼ 0:01. 1 P cos n is divergent. 9. (a) Show that the series (b) Show that the series n¼1 1 P n¼1 ðcos nÞ=n2 is convergent. 1 P ð1Þn pffiffi is 10. Use an argument similar to that in Example 3.7.6(f) to show that the series n n¼1 convergent. P P 2 11. If an with an > 0 is convergent, then is an always convergent? Either prove it or give a counterexample. P pffiffiffiffiffi P 12. If an with an > 0 is convergent, then is an always convergent? Either prove it or give a counterexample. P pffiffiffiffiffiffiffiffiffiffiffiffiffi P 13. If an with an > 0 is convergent, then is an anþ1 always convergent? Either prove it or give a counterexample. P 14. P If an with an > 0 is convergent, and if bn :¼ ða1 þ þ an Þ=n for n 2 N; then show that bn is always divergent. 1 P 15. Let aðnÞ be such that ðaðnÞÞ is a decreasing sequence of strictly positive numbers. If sðnÞ n¼1 denotes the nth partial sum, show (by grouping the terms in sð2n Þ in two different ways) that n n n n1 1 a 2n1 þ að2n Þ. 2 ðað1Þ þ 2að2Þ þ þ 2 að2 ÞÞ sð2 Þ að1Þ þ 2að2Þ þ þ 2 1 1 P P Use these inequalities to show that aðnÞ converges if and only if 2n að2n Þ converges. This n¼1 n¼1 result is often called the Cauchy Condensation Test; it is very powerful. 1 P ð1=np Þ for p > 0. 16. Use the Cauchy Condensation Test to discuss the p-series n¼1 17. Use the Cauchy Condensation Test to establish the divergence of the series: X X 1 1 , , (b) (a) n ln n n ð ln n Þ ðln ln nÞ X 1 . (c) nðln nÞðln ln nÞðln ln ln nÞ 18. Show that if c > 1, then the following series are convergent: (a) X 1 , nðln nÞc (b) X 1 . nðln nÞðln ln nÞc C04 12/09/2010 14:49:5 Page 102 CHAPTER 4 LIMITS ‘‘Mathematical analysis’’ is generally understood to refer to that area of mathematics in which systematic use is made of various limiting concepts. In the preceding chapter we studied one of these basic limiting concepts: the limit of a sequence of real numbers. In this chapter we will encounter the notion of the limit of a function. The rudimentary notion of a limiting process emerged in the 1680s as Isaac Newton (1642–1727) and Gottfried Leibniz (1646–1716) struggled with the creation of the Calculus. Though each person’s work was initially unknown to the other and their creative insights were quite different, both realized the need to formulate a notion of function and the idea of quantities being ‘‘close to’’ one another. Newton used the word ‘‘fluent’’ to denote a relationship between variables, and in his major work Principia in 1687 he discussed limits ‘‘to which they approach nearer than by any given difference, but never go beyond, nor in effect attain to, till the quantities are diminished in infinitum.’’ Leibniz introduced the term ‘‘function’’ to indicate a quantity that depended on a variable, and he invented ‘‘infinitesimally small’’ numbers as a way of handling the concept of a limit. The term ‘‘function’’ soon became standard terminology, and Leibniz also introduced the term ‘‘calculus’’ for this new method of calculation. In 1748, Leonhard Euler (1707–1783) published his two-volume treatise Introductio in Analysin Infinitorum, in which he discussed power series, the exponential and logarithmic functions, the trigonometric functions, and many related topics. This was followed by Institutiones Calculi Differentialis in 1755 and the three-volume Institutiones Calculi Integralis in 1768–1770. These works remained the standard textbooks on calculus for many years. But the concept of limit was very intuitive and its looseness led to a number of problems. Verbal descriptions of the limit concept were proposed by other mathematicians of the era, but none was adequate to provide the basis for rigorous proofs. In 1821, Augustin-Louis Cauchy (1789–1857) published his lectures on analysis in his Cours d’Analyse, which set the standard for mathematical exposition for many years. He was concerned with rigor and in many ways raised the level of precision in mathematical discourse. He formulated definitions and presented arguments with greater care than his predecessors, but the concept of limit still remained elusive. In an early chapter he gave the following definition: If the successive values attributed to the same variable approach indefinitely a fixed value, such that they finally differ from it by as little as one wishes, this latter is called the limit of all the others. The final steps in formulating a precise definition of limit were taken by Karl Weierstrass (1815–1897). He insisted on precise language and rigorous proofs, and his definition of limit is the one we use today. 102 C04 12/09/2010 14:49:6 Page 103 4.1 LIMITS OF FUNCTIONS 103 Gottfried Leibniz Gottfried Wilhelm Leibniz (1646–1716) was born in Leipzig, Germany. He was six years old when his father, a professor of philosophy, died and left his son the key to his library and a life of books and learning. Leibniz entered the University of Leipzig at age 15, graduated at age 17, and received a Doctor of Law degree from the University of Altdorf four years later. He wrote on legal matters, but was more interested in philosophy. He also developed original theories about language and the nature of the universe. In 1672, he went to Paris as a diplomat for four years. While there he began to study mathematics with the Dutch mathematician Christiaan Huygens. His travels to London to visit the Royal Academy further stimulated his interest in mathematics. His background in philosophy led him to very original, though not always rigorous, results. Unaware of Newtons’s unpublished work, Leibniz published papers in the 1680s that presented a method of finding areas that is known today as the Fundamental Theorem of Calculus. He coined the term ‘‘calculus’’ and invented the dy=dx and elongated S notations that are used today. Unfortunately, some followers of Newton accused Leibniz of plagiarism, resulting in a dispute that lasted until Leibniz’s death. Their approaches to calculus were quite different and it is now evident that their discoveries were made independently. Leibniz is now renowned for his work in philosophy, but his mathematical fame rests on his creation of the calculus. Section 4.1 Limits of Functions In this section we will introduce the important notion of the limit of a function. The intuitive idea of the function f having a limit L at the point c is that the values f (x) are close to L when x is close to (but different from) c. But it is necessary to have a technical way of working with the idea of ‘‘close to’’ and this is accomplished in the e-d definition given below. In order for the idea of the limit of a function f at a point c to be meaningful, it is necessary that f be defined at points near c. It need not be defined at the point c, but it should be defined at enough points close to c to make the study interesting. This is the reason for the following definition. 4.1.1 Definition Let A R. A point c 2 R is a cluster point of A if for every d > 0 there exists at least one point x 2 A; x 6¼ c such that jx cj < d. This definition is rephrased in the language of neighborhoods as follows: A point c is a cluster point of the set A if every d-neighborhood V d ðcÞ ¼ ðc d; c þ dÞ of c contains at least one point of A distinct from c. Note The point c may or may not be a member of A, but even if it is in A, it is ignored when deciding whether it is a cluster point of A or not, since we explicitly require that there be points in V d ðcÞ \ A distinct from c in order for c to be a cluster point of A. For example, if A :¼ {1, 2}, then the point 1 is not a cluster point of A, since choosing d :¼ 12 gives a neighborhood of 1 that contains no points of A distinct from 1. The same is true for the point 2, so we see that A has no cluster points. C04 12/09/2010 14:49:6 104 Page 104 CHAPTER 4 LIMITS 4.1.2 Theorem A number c 2 R is a cluster point of a subset A of R if and only if there exists a sequence (an) in A such that limðan Þ ¼ c and an 6¼ c for all n 2 N. Proof. If c is a cluster point of A, then for any n 2 N the (l=n)-neighborhood V 1=n ðcÞ contains at least one point an in A distinct from c. Then an 2 A; an 6¼ c, and jan cj < 1=n implies limðan Þ ¼ c. Conversely, if there exists a sequence (an) in Anfcg with limðan Þ ¼ c, then for any d > 0 there exists K such that if n K, then an 2 V d ðcÞ. Therefore the d-neighborhood V d ðcÞ of c contains the points an, for n K, which belong to A and are distinct from c. Q.E.D. The next examples emphasize that a cluster point of a set may or may not belong to the set. 4.1.3 Examples (a) For the open interval A1 :¼ ð0; 1Þ, every point of the closed interval [0,1] is a cluster point of A1. Note that the points 0, 1 are cluster points of A1, but do not belong to A1. All the points of A1 are cluster points of A1. (b) A finite set has no cluster points. (c) The infinite set N has no cluster points. (d) The set A4 :¼ f1=n : n 2 N g has only the point 0 as a cluster point. None of the points in A4 is a cluster point of A4. (e) If I :¼ ½0; 1, then the set A5 :¼ I \ Q consists of all the rational numbers in I. It follows & from the Density Theorem 2.4.8 that every point in I is a cluster point of A5. Having made this brief detour, we now return to the concept of the limit of a function at a cluster point of its domain. The Definition of the Limit We now state the precise definition of the limit of a function f at a point c. It is important to note that in this definition, it is immaterial whether f is defined at c or not. In any case, we exclude c from consideration in the determination of the limit. 4.1.4 Definition Let A R, and let c be a cluster point of A. For a function f : A ! R, a real number L is said to be a limit of f at c if, given any e > 0, there exists a d > 0 such that if x 2 A and 0 < jx cj < d, then j f ðxÞ Lj < e. Remarks (a) Since the value of d usually depends on e, we will sometimes write dðeÞ instead of d to emphasize this dependence. (b) The inequality 0 < jx cj is equivalent to saying x 6¼ c. If L is a limit of f at c, then we also say that f converges to L at c. We often write L ¼ lim f ðxÞ or x!c L ¼ lim f : x!c We also say that ‘‘f (x) approaches L as x approaches c.’’ (But it should be noted that the points do not actually move anywhere.) The symbolism f ðxÞ ! L as x!c is also used sometimes to express the fact that f has limit L at c. C04 12/09/2010 14:49:6 Page 105 4.1 LIMITS OF FUNCTIONS 105 If the limit of f at c does not exist, we say that f diverges at c. Our first result is that the value L of the limit is uniquely determined. This uniqueness is not part of the definition of limit, but must be deduced. 4.1.5 Theorem If f : A ! R and if c is a cluster point of A, then f can have only one limit at c. Proof. Suppose that numbers L and L0 satisfy Definition 4.1.4. For any e > 0, there exists dðe=2Þ > 0 such that if x 2 A and 0 < jx cj < dðe=2Þ, then j f ðxÞ Lj < e=2. Also there exists d0 ðe=2Þ such that if x 2 A and 0 < jx cj < d0 ðe=2Þ, then j f ðxÞ L0 j < e=2. Now let d :¼ inf fdðe=2Þ; d0 ðe=2Þg. Then if x 2 A and 0 < jx cj < d, the Triangle Inequality implies that jL L0 j L f ðxÞ þ j f ðxÞ L0 j < e=2 þ e=2 ¼ e: Since e > 0 is arbitrary, we conclude that L L0 ¼ 0, so that L ¼ L0 . Q.E.D. The definition of limit can be very nicely described in terms of neighborhoods. (See Figure 4.1.1.) We observe that because V d ðcÞ ¼ ðc d; c þ dÞ ¼ fx : jx cj < dg; the inequality 0 < jx cj < d is equivalent to saying that x 6¼ c and x belongs to the d-neighborhood V d ðcÞ of c. Similarly, the inequality j f ðxÞ Lj < e is equivalent to saying that f (x) belongs to the e-neighborhood V e ðLÞ of L. In this way, we obtain the following result. The reader should write out a detailed argument to establish the theorem. Figure 4.1.1 The limit of f at c is L 4.1.6 Theorem Let f : A ! R and let c be a cluster point of A. Then the following statements are equivalent. (i) lim f ðxÞ ¼ L. x!c (ii) Given any e-neighborhood V e ðLÞ of L, there exists a d-neighborhood V d ðcÞ of c such that if x 6¼ c is any point in V d ðcÞ \ A, then f (x) belongs to V e ðLÞ. We now give some examples that illustrate how the definition of limit is applied. 4.1.7 Examples (a) lim b ¼ b. x!c To be more explicit, let f ðxÞ :¼ b for all x 2 R. We want to show that lim f ðxÞ ¼ b. If x!c e > 0 is given, we let d :¼ 1. (In fact, any strictly positive d will serve the purpose.) Then if C04 12/09/2010 14:49:7 106 Page 106 CHAPTER 4 LIMITS 0 < jx cj < 1, we have j f ðxÞ bj ¼ jb bj ¼ 0 < e. Since e > 0 is arbitrary, we conclude from Definition 4.1.4 that lim f ðxÞ ¼ b. x!c (b) lim x ¼ c. x!c Let gðxÞ :¼ x for all x 2 R. If e > 0, we choose dðeÞ :¼ e. Then if 0 < jx cj < dðeÞ, we have jgðxÞ cj ¼ jx cj < e. Since e > 0 is arbitrary, we deduce that lim g ¼ c. x!c (c) lim x2 ¼ c2 : x!c Let hðxÞ :¼ x2 for all x 2 R. We want to make the difference hðxÞ c2 ¼ x2 c2 less than a preassigned e > 0 by taking x sufficiently close to c. To do so, we note that x2 c2 ¼ ðx þ cÞðx cÞ. Moreover, if jx cj < 1, then jxj < jcj þ 1 so that jx þ cj jxj þ jcj < 2 jcj þ 1: Therefore, if jx cj < 1, we have 2 x c2 ¼ jx þ cjjx cj < ð2 jcj þ 1Þ jx cj: ð1Þ Moreover this last term will be less than e provided we take jx cj < e=ð2jcj þ 1Þ. Consequently, if we choose e dðeÞ :¼ inf 1; ; 2j c j þ 1 then if 0 < jx cj < dðeÞ, it will follow first that jx cj < 1 so that (1) is valid, and therefore, since jx cj < e=ð2jcj þ 1Þ that 2 x c2 < ð2jcj þ 1Þjx cj < e: Since we have a way of choosing dðeÞ > 0 for an arbitrary choice of e > 0, we infer that lim hðxÞ ¼ lim x2 ¼ c2 . x!c x!c 1 1 (d) lim ¼ if c > 0. x!c x c Let wðxÞ :¼ 1=x for x > 0 and let c > 0. To show that lim w ¼ 1=c we wish to make x!c the difference wðxÞ 1 ¼ 1 1 c x c less than a preassigned e > 0 by taking x sufficiently close to c > 0. We first note that 1 1 1 ¼ ðc xÞ ¼ 1 jx cj x c cx cx for x > 0. It is useful to get an upper bound for the term 1=(cx) that holds in some neighborhood of c. In particular, if jx cj < 12 c, then 12 c < x < 32 c (why?), so that 1 2 1 < for jx cj < c: cx c2 2 Therefore, for these values of x we have 1 2 ð2Þ wðxÞ c c2 jx cj: 0< C04 12/09/2010 14:49:7 Page 107 4.1 LIMITS OF FUNCTIONS 107 In order to make this last term less than e it suffices to take jx cj < 12 c2 e. Consequently, if we choose 1 1 2 dðeÞ :¼ inf c; c e ; 2 2 1 then if 0 < jx cj < dðeÞ, it will follow first that jx cj < 2 c so that (2) is valid, and 1 2 therefore, since jx cj < 2 c e, that wðxÞ 1 ¼ 1 1 < e: c x c Since we have a way of choosing dðeÞ > 0 for an arbitrary choice of e > 0, we infer that lim w ¼ 1=c. x!c x3 4 4 ¼ : x!2 x2 þ 1 5 Let cðxÞ :¼ ðx3 4Þ=ðx2 þ 1Þ for x 2 R. Then a little algebraic manipulation gives us 4 5x3 4x2 24 cðxÞ ¼ 5 5 ð x2 þ 1 Þ 3 5x þ 6x þ 12 jx 2j: ¼ 5 ð x2 þ 1 Þ (e) lim To get a bound on the coefficient of jx 2j, we restrict x by the condition 1 < x < 3. For x in this interval, we have 5x2 þ 6x þ 12 5 32 þ 6 3 þ 12 ¼ 75 and 5ðx2 þ 1Þ 5ð1 þ 1Þ ¼ 10, so that cðxÞ 4 75 jx 2j ¼ 15 jx 2j: 5 10 2 Now for given e > 0, we choose 2 e : dðeÞ :¼ inf 1; 15 Then if 0 < jx 2j < dðeÞ, we have jcðxÞ ð4=5Þj ð15=2Þjx 2j < e. Since e > 0 is & arbitrary, the assertion is proved. Sequential Criterion for Limits The following important formulation of limit of a function is in terms of limits of sequences. This characterization permits the theory of Chapter 3 to be applied to the study of limits of functions. 4.1.8 Theorem (Sequential Criterion) Let f : A ! R and let c be a cluster point of A. Then the following are equivalent. (i) lim f ¼ L: x!c (ii) For every sequence (xn) in A that converges to c such that xn 6¼ c for all n 2 N, the sequence ( f(xn)) converges to L. Proof. (i) ) (ii). Assume f has limit L at c, and suppose (xn) is a sequence in A with limðxn Þ ¼ c and xn 6¼ c for all n. We must prove that the sequence ð f ðxn ÞÞ converges to L. Let e > 0 be given. Then by Definition 4.1.4, there exists d > 0 such that if x 2 A satisfies C04 12/09/2010 14:49:8 108 Page 108 CHAPTER 4 LIMITS 0 < jx cj < d, then f (x) satisfies j f ðxÞ Lj < e. We now apply the definition of convergent sequence for the given d to obtain a natural number K(d) such that if n > K ðdÞ then jxn cj < d. But for each such xn we have j f ðxn Þ Lj < e. Thus if n > K ðdÞ, then j f ðxn Þ Lj < e. Therefore, the sequence ð f ðxn ÞÞ converges to L. (ii) ) (i). [The proof is a contrapositive argument.] If (i) is not true, then there exists an e0 -neighborhood V e0 ðLÞ such that no matter what d-neighborhood of c we pick, there will be at least one number xd in A \ V d ðcÞ with xd 6¼ c such that f ðxd Þ 2 = V e0 ðLÞ. Hence for every n 2 N, the (1=n)-neighborhood of c contains a number xn such that 0 < jxn cj < 1=n and xn 2 A ; for all n2N: but such that j f ð xn Þ L j e 0 We conclude that the sequence (xn) in An{c} converges to c, but the sequence ð f ðxn ÞÞ does not converge to L. Therefore we have shown that if (i) is not true, then (ii) is not true. We Q.E.D. conclude that (ii) implies (i). We shall see in the next section that many of the basic limit properties of functions can be established by using corresponding properties for convergent sequences. For example, we know from our with sequences that if (xn) is any sequence that converges to a work number c, then x2n converges to c2. Therefore, by the sequential criterion, we can conclude that the function hðxÞ :¼ x2 has limit lim hðxÞ ¼ c2 . x!c Divergence Criteria It is often important to be able to show (i) that a certain number is not the limit of a function at a point, or (ii) that the function does not have a limit at a point. The following result is a consequence of (the proof of) Theorem 4.1.8. We leave the details of its proof as an important exercise. 4.1.9 Divergence Criteria Let A R, let f : A ! R and let c 2 R be a cluster point of A. (a) If L 2 R, then f does not have limit L at c if and only if there exists a sequence (xn) in A with xn 6¼ c for all n 2 N such that the sequence (xn) converges to c but the sequence ð f ðxn ÞÞ does not converge to L. (b) The function f does not have a limit at c if and only if there exists a sequence (xn) in A with xn 6¼ c for all n 2 N such that the sequence (xn) converges to c but the sequence ð f ðxn ÞÞ does not converge in R. We now give some applications of this result to show how it can be used. 4.1.10 Examples (a) lim ð1=xÞ does not exist in R. x!0 As in Example 4.1.7(d), let wðxÞ :¼ 1=x for x > 0. However, here we consider c ¼ 0. The argument given in Example 4.1.7(d) breaks down if c ¼ 0 since we cannot obtain a bound such as that in (2) of that example. Indeed, if we take the sequence (xn) with xn :¼ 1=n for n 2 N, then limðxn Þ ¼ 0, but wðxn Þ ¼ 1=ð1=nÞ ¼ n. As we know, the sequence ðwðxn ÞÞ ¼ ðnÞ is not convergent in R, since it is not bounded. Hence, by Theorem 4.1.9(b), lim ð1=xÞ does not exist in R. x!0 (b) lim sgnðxÞ does not exist. x!0 C04 12/09/2010 14:49:9 Page 109 4.1 LIMITS OF FUNCTIONS Let the signum function sgn be defined by 8 for < þ1 sgnðxÞ :¼ 0 for : 1 for 109 x > 0; x ¼ 0; x < 0: Note that sgnðxÞ ¼ x=jxj for x 6¼ 0. (See Figure 4.1.2.) We shall show that sgn does not have a limit at x ¼ 0. We shall do this by showing that there is a sequence (xn) such that limðxn Þ ¼ 0, but such that ðsgnðxn ÞÞ does not converge. Figure 4.1.2 The signum function Indeed, let xn :¼ ð1Þn =n for n 2 N so that limðxn Þ ¼ 0. However, since sgnðxn Þ ¼ ð1Þn for n2N; it follows from Example 3.4.6(a) that ðsgnðxn ÞÞ does not converge. Therefore lim sgnðxÞ x!0 does not exist. (c)y lim sinð1=xÞ does not exist in R. x!0 Let gðxÞ :¼ sinð1=xÞ for x 6¼ 0. (See Figure 4.1.3.) We shall show that g does not have a limit at c ¼ 0, by exhibiting two sequences (xn) and (yn) with xn 6¼ 0 and yn 6¼ 0 for all n 2 N and such that limðxn Þ ¼ 0 and limðyn Þ ¼ 0, but such that limðgðxn ÞÞ 6¼ limðgðyn ÞÞ. In view of Theorem 4.1.9 this implies that lim g cannot exist. (Explain why.) x!0 Figure 4.1.3 The function gðxÞ ¼ sinð1=xÞðx 6¼ 0Þ Indeed, we recall from calculus that sin t ¼ 0 if t ¼ np for n 2 Z, and that sin t ¼ þ1 if t ¼ 12 p þ 2p n for n 2 Z. Now let xn :¼ 1=np for n 2 N; then limðxn Þ ¼ 0 and gðxn Þ ¼ np ¼ 0 for all n 2 N, so that limðgðxn ÞÞ ¼ 0. On the other hand, sin let yn :¼ 1 1 1 p þ 2pn for n 2 N; then lim ð y Þ ¼ 0 and g ð y Þ ¼ sin p þ 2p n ¼ 1 for all n n 2 2 & n 2 N, so that limðgðyn ÞÞ ¼ 1. We conclude that lim sinð1=xÞ does not exist. x!0 y In order to have some interesting applications in this and later examples, we shall make use of well-known properties of trigonometric and exponential functions that will be established in Chapter 8. C04 12/09/2010 14:49:10 110 Page 110 CHAPTER 4 LIMITS Exercises for Section 4.1 1. Determine acondition on jx 1j that will assure that: (a) x2 1 < 12, (b) x2 1 < 1=103 , 2 3 for a given n 2 N , (d) x 1 < 1=n for a given n 2 N . (c) x 1 < 1=n 2. Determine a condition on jx 4j that will assure that: pffiffiffi pffiffiffi (a) j x 2j < 12, (b) j x 2j < 102 . 3. Let c be a cluster point of A R and let f : A ! R. Prove that lim f ðxÞ ¼ L if and only if x!c lim j f ðxÞ Lj ¼ 0. x!c 4. Let f :¼ R ! R and let c 2 R. Show that lim f ðxÞ ¼ L if and only if lim f ðx þ cÞ ¼ L. x!c x!0 2 5. Let I :¼ ð20; aÞ where a > 0, and let gðxÞ :¼ x for x 2 I. For any points x, c 2 I, show that gðxÞ c 2ajx cj. Use this inequality to prove that lim x2 ¼ c2 for any c 2 I. x!c 6. Let I be an interval in R, let f : I ! R, and let c 2 I. Suppose there exist constants K and L such that j f ðxÞ Lj K jx cj for x 2 I. Show that lim f ðxÞ ¼ L. x!c 7. Show that lim x3 ¼ c3 for any c 2 R. x!c pffiffiffi pffiffiffi 8. Show that lim x ¼ c for any c > 0. x!c 9. Use either the e-d definition of limit or the Sequential Criterion for limits, to establish the following limits. 1 x 1 ¼ 1, (b) lim ¼ , (a) lim x!2 1 x x!1 1 þ x 2 x2 x2 x þ 1 1 ¼ . ¼ 0, (d) lim (c) lim x!0 jxj x!1 xþ1 2 10. Use the definition of limit to show that (a) lim x2 þ 4x ¼ 12, (b) x!2 11. Use the definition of limit to prove the following. 2x þ 3 (a) lim ¼ 3, (b) x!3 4x 9 12. Show that the following limits do not exist. 1 (b) (a) lim 2 ðx > 0Þ , x!0 x (d) (c) lim ðx þ sgnðxÞÞ, x!0 lim xþ5 ¼ 4. þ3 x!1 2x x2 3x ¼ 2: x!6 x þ 3 lim 1 lim pffiffiffi ðx > 0Þ , x lim sin 1=x2 . x!0 x!0 13. Suppose the function f : R ! R has limit L at 0, and let a > 0. If g : R ! R is defined by gðxÞ :¼ f ðaxÞ for x 2 R, show that lim gðxÞ ¼ L. x!0 14. Let c 2 R and let f : R ! R be such that lim ð f ðxÞÞ2 ¼ L. x!c (a) Show that if L ¼ 0, then lim f ðxÞ ¼ 0. x!c (b) Show by example that if L 6¼ 0, then f may not have a limit at c. 15. Let f : R ! R be defined by setting f ðxÞ :¼ x if x is rational, and f ðxÞ ¼ 0 if x is irrational. (a) Show that f has a limit at x ¼ 0. (b) Use a sequential argument to show that if c 6¼ 0, then f does not have a limit at c. 16. Let f : R ! R, let I be an open interval in R, and let c 2 I. If f1 is the restriction of f to I, show that f1 has a limit at c if and only if f has a limit at c, and that the limits are equal. 17. Let f : R ! R, let J be a closed interval in R, and let c 2 J. If f2 is the restriction of f to J, show that if f has a limit at c then f2 has a limit at c. Show by example that it does not follow that if f2 has a limit at c, then f has a limit at c. C04 12/09/2010 14:49:10 Page 111 4.2 LIMIT THEOREMS 111 Section 4.2 Limit Theorems We shall now obtain results that are useful in calculating limits of functions. These results are parallel to the limit theorems established in Section 3.2 for sequences. In fact, in most cases these results can be proved by using Theorem 4.1.8 and results from Section 3.2. Alternatively, the results in this section can be proved by using e-d arguments that are very similar to the ones employed in Section 3.2. 4.2.1 Definition Let A R, let f : A ! R, and let c 2 R be a cluster point of A. We say that f is bounded on a neighborhood of c if there exists a d-neighborhood V d ðcÞ of c and a constant M > 0 such that we have j f ðxÞj M for all x 2 A \ V d ðcÞ. 4.2.2 Theorem If A R and f : A ! R has a limit at c 2 R, then f is bounded on some neighborhood of c. Proof. If L :¼ lim f , then for e ¼ 1, there exists d > 0 such that if 0 < jx cj < d, then x!c j f ðxÞ Lj < 1; hence (by Corollary 2.2.4(a)), j f ðxÞj jLj j f ðxÞ Lj < 1: Therefore, if x 2 A \ V d ðcÞ; x 6¼ c, then j f ðxÞj jLj þ 1. If c 2 = A, we take M ¼ jLj þ 1, while if c 2 A we take M :¼ supfj f ðcÞj; jLj þ 1g. It follows that if x 2 A \ V d ðcÞ, then j f ðxÞj M. This shows that f is bounded on the neighborhood V d ðcÞ of c. Q.E.D. The next definition is similar to the definition for sums, differences, products, and quotients of sequences given in Section 3.2. 4.2.3 Definition Let A R and let f and g be functions defined on A to R. We define the sum f þ g, the difference f g, and the product fg on A to R to be the functions given by ð f þ gÞðxÞ :¼ f ðxÞ þ gðxÞ; ð f gÞðxÞ :¼ f ðxÞ gðxÞ; ð f gÞðxÞ :¼ f ðxÞgðxÞ for all x 2 A. Further, if b 2 R, we define the multiple bf to be the function given by ðbf ÞðxÞ :¼ bf ðxÞ for all x 2 A : Finally, if hðxÞ 6¼ 0 for x 2 A, we define the quotient f=h to be the function given by f f ðxÞ for all x 2 A: ðxÞ :¼ h hðxÞ 4.2.4 Theorem Let A R, let f and g be functions on A to R, and let c 2 R be a cluster point of A. Further, let b 2 R. (a) If lim f ¼ L and lim g ¼ M, then: x!c x!c lim ð f þ gÞ ¼ L þ M; x!c lim ð f gÞ ¼ LM; x!c lim ð f gÞ ¼ L M; x!c lim ðbf Þ ¼ bL: x!c (b) If h : A ! R, if hðxÞ 6¼ 0 for all x 2 A, and if lim h ¼ H 6¼ 0, then x!c f L ¼ : lim x!c h H C04 12/09/2010 14:49:11 112 Page 112 CHAPTER 4 LIMITS Proof. One proof of this theorem is exactly similar to that of Theorem 3.2.3. Alternatively, it can be proved by making use of Theorems 3.2.3 and 4.1.8. For example, let (xn) be any sequence in A such that xn 6¼ c for n 2 N, and c ¼ limðxn Þ. It follows from Theorem 4.1.8 that limð f ðxn ÞÞ ¼ L; limðgðxn ÞÞ ¼ M: On the other hand, Definition 4.2.3 implies that ð f gÞðxn Þ ¼ f ðxn Þgðxn Þ for n 2 N: Therefore an application of Theorem 3.2.3 yields limðð f gÞðxn ÞÞ ¼ limð f ðxn Þgðxn ÞÞ ¼ ½limð f ðxn ÞÞ ½limðgðxn ÞÞ ¼ LM: Consequently, it follows from Theorem 4.1.8 that lim ð f gÞ ¼ limðð f gÞðxn ÞÞ ¼ LM: x!c The other parts of this theorem are proved in a similar manner. We leave the details to the reader. Q.E.D. Remark Let A R, and let f 1 ; f 2 ; . . . ; f n be functions on A to R, and let c be a cluster point of A. If Lk :¼ lim f k for k ¼ 1; . . . ; n; then it follows from Theorem 4.2.4 by an x!c Induction argument that L1 þ L2 þ þ Ln ¼ lim ð f 1 þ f 2 þ þ f n Þ; x!c and L1 L2 Ln ¼ limð f 1 f 2 f n Þ: In particular, we deduce that if L ¼ lim f and n 2 N, then x!c Ln ¼ lim ð f ðxÞÞn : x!c 4.2.5 Examples (a) Some of the limits that were established in Section 4.1 can be proved by using Theorem 4.2.4. For example, it follows from this result that since lim x ¼ c, then lim x2 ¼ c2 , and that if c > 0, then x!c x!c lim 1 x!c x ¼ 1 1 ¼ : lim x c x!c (b) lim x2 þ 1 x3 4 ¼ 20. x!2 It follows from Theorem 4.2.4 that lim x þ 1 x 4 ¼ 2 x!2 (c) lim x!2 3 lim x þ 1 2 x!2 ¼ 5 4 ¼ 20: x3 4 x2 þ 1 4 ¼ . 5 lim x3 4 x!2 C04 12/09/2010 14:49:11 Page 113 4.2 LIMIT THEOREMS 113 If we apply Theorem 4.2.4(b), we have lim x3 4 x3 4 x!2 4 2 ¼ : ¼ lim 2 x!2 x þ 1 5 lim x þ 1 x!2 2 Note that since the limit in the denominator [i.e., lim x þ 1 ¼ 5] is not equal to 0, then x!2 Theorem 4.2.4(b) is applicable. x2 4 4 ¼ . (d) lim ¼ x!2 3x 6 3 If we let f ðxÞ :¼ x2 4 and hðxÞ :¼ 3x 6 for x 2 R, then we cannot use Theorem 4.2.4(b) to evaluate lim ð f ðxÞ=hðxÞÞ because x!2 H ¼ lim hðxÞ ¼ lim ð3x 6Þ ¼ 3 2 6 ¼ 0: x!2 x!2 However, if x 6¼ 2, then it follows that x 2 4 ð x þ 2Þ ð x 2Þ 1 ¼ ¼ ðx þ 2Þ: 3x 6 3ð x 2Þ 3 Therefore we have x2 4 1 1 ¼ lim ðx þ 2Þ ¼ lim x þ 2 lim x!2 3x 6 x!2 3 3 x!2 4 ¼ : 3 Note that the function gðxÞ ¼ ðx2 4Þ=ð3x 6Þ has a limit at x ¼ 2 even though it is not defined there. 1 (e) lim does not exist in R. x!0 x Of course lim 1 ¼ 1 and H :¼ lim x ¼ 0. However, since H ¼ 0, we cannot use x!0 x!0 Theorem 4.2.4(b) to evaluate lim ð1=xÞ. In fact, as was seen in Example 4.1.10(a), the x!0 function wðxÞ ¼ 1=x does not have a limit at x ¼ 0. This conclusion also follows from Theorem 4.2.2 since the function wðxÞ ¼ 1=x is not bounded on a neighborhood of x ¼ 0. (f) If p is a polynomial function, then lim pðxÞ ¼ pðcÞ. x!c Let p be a polynomial function on R so that pðxÞ ¼ an xn þ an1 xn1 þ þ a1 x þ a0 for all x 2 R. It follows from Theorem 4.2.4 and the fact that lim xk ¼ ck that x!c lim pðxÞ ¼ lim an xn þ an1 xn1 þ þ a1 x þ a0 x!c x!c ¼ lim ðan xn Þ þ lim an1 xn1 þ þ lim ða1 xÞ þ lim a0 x!c x!c x!c x!c ¼ an cn þ an1 cn1 þ þ a1 c þ a0 ¼ pðcÞ: Hence lim pðxÞ ¼ pðcÞ for any polynomial function p. x!c (g) If p and q are polynomial functions on R and if qðcÞ 6¼ 0, then pðxÞ pðcÞ ¼ : lim x!c qðxÞ qð c Þ Since q(x) is a polynomial function, it follows from a theorem in algebra that there are at most a finite number of real numbers a1 ; . . . ; am [the real zeroes of q(x)] such that q aj ¼ 0 and such that if x 2 = fa1 ; . . . ; am g, then qðxÞ 6¼ 0. Hence, if x 2 = fa1 ; . . . ; am g, we can define pðxÞ : rðxÞ :¼ qðxÞ C04 12/09/2010 14:49:11 114 Page 114 CHAPTER 4 LIMITS If c is not a zero of q(x), then qðcÞ 6¼ 0, and it follows from part (f) that lim qðxÞ ¼ qðcÞ 6¼ 0. x!c Therefore we can apply Theorem 4.2.4(b) to conclude that lim pðxÞ pðcÞ pðxÞ x!c ¼ : ¼ x!c qðxÞ lim qðxÞ qðcÞ & lim x!c The next result is a direct analogue of Theorem 3.2.6. 4.2.6 Theorem Let A R, let f : A ! R, and let c 2 R be a cluster point of A. If a f ðxÞ b f or all x 2 A; x 6¼ c; and if lim f exists, then a lim f b. x!c x!c Proof. Indeed, if L ¼ lim f , then it follows from Theorem 4.1.8 that if (xn) is any x!c sequence of real numbers such that c 6¼ xn 2 A for all n 2 N and if the sequence (xn) converges to c, then the sequence ð f ðxn ÞÞ converges to L. Since a f ðxn Þ b for all n 2 N, it follows from Theorem 3.2.6 that a L b. Q.E.D. We now state an analogue of the Squeeze Theorem 3.2.7. We leave its proof to the reader. 4.2.7 Squeeze Theorem Let A R, let f, g, h: A ! R, and let c 2 R be a cluster point of A. If f ðxÞ gðxÞ hðxÞ f or all x 2 A; x 6¼ c; and if lim f ¼ L ¼ lim h, then lim g ¼ L. x!c x!c x!c 4.2.8 Examples (a) lim x3=2 ¼ 0 ðx > 0Þ. x!0 Let f ðxÞ :¼ x for x > 0. Since the inequality x < x1=2 1 holds for 0 < x 1 (why?), it follows that x2 f ðxÞ ¼ x3=2 x for 0 < x 1. Since 3=2 lim x2 ¼ 0 and x!0 lim x ¼ 0; x!0 it follows from the Squeeze Theorem 4.2.7 that lim x3=2 ¼ 0. x!0 (b) lim sin x ¼ 0. x!0 It will be proved later (see Theorem 8.4.8), that x sin x x for all x 0: Since lim ðxÞ ¼ 0, it follows from the Squeeze Theorem that lim sin x ¼ 0. x!0 x!0 (c) lim cos x ¼ 1. x!0 It will be proved later (see Theorem 8.4.8) that ð1Þ 1 12 x2 cos x 1 for all x 2 R: Since lim 1 12 x2 ¼ 1, it follows from the Squeeze Theorem that lim cos x ¼ 1. x!0 x!0 cos x 1 (d) lim ¼ 0. x!0 x C04 12/09/2010 14:49:12 Page 115 4.2 LIMIT THEOREMS 115 We cannot use Theorem 4.2.4(b) to evaluate this limit. (Why not?) However, it follows from the inequality (1) in part (c) that 12 x ðcos x 1Þ=x 0 for x>0 0 ðcos x 1Þ=x 12 x for x < 0: and that Now let f ðxÞ :¼ x=2 for x 0 and f ðxÞ :¼ 0 for x < 0, and let hðxÞ :¼ 0 for x 0 and hðxÞ :¼ x=2 for x < 0. Then we have f ðxÞ ðcos x 1Þ=x hðxÞ for x 6¼ 0: Since it is readily seen that lim f ¼ 0 ¼ lim h, it follows from the Squeeze Theorem that x!0 x!0 lim ðcos x 1Þ=x ¼ 0. x!0 sin x (e) lim ¼ 1. x!0 x Again we cannot use Theorem 4.2.4(b) to evaluate this limit. However, it will be proved later (see Theorem 8.4.8) that x 16 x3 sin x x for x 0 and that x sin x x 16 x3 for x 0: Therefore it follows (why?) that 1 16 x2 ðsin xÞ=x 1 for all x 6¼ 0: But since lim 1 16 x2 ¼ 1 16 lim x2 ¼ 1, we infer from the Squeeze Theorem that x!0 x!0 lim ðsin xÞ=x ¼ 1. x!0 (f) lim ðx sinð1=xÞÞ ¼ 0. x!0 Let f ðxÞ ¼ x sinð1=xÞ for x ¼ 6 0. Since 1 sin z 1 for all z 2 R, we have the inequality jxj f ðxÞ ¼ x sinð1=xÞ jxj for all x 2 R, x 6¼ 0. Since lim jxj ¼ 0, it follows from the Squeeze Theorem that lim f ¼ 0. x!0 x!0 For a graph, see Figure 5.1.3 or the cover of this book. & There are results that are parallel to Theorems 3.2.9 and 3.2.10; however, we will leave them as exercises. We conclude this section with a result that is, in some sense, a partial converse to Theorem 4.2.6. 4.2.9 Theorem Let A R, let f : A ! R and let c 2 R be a cluster point of A. If h i lim f > 0 respectively; lim f < 0 ; x!c x!c then there exists a neighborhood Vd(c) of c such that f (x) > 0 [respectively, f (x) < 0] for all x 2 A \ V d ðcÞ, x 6¼ c. Proof. Let L :¼ lim f and suppose that L > 0. We take e ¼ 12 L > 0 in Definition 4.1.4, and x!c obtain a number d > 0 such that if 0 < jx cj < d and x 2 A, then j f ðxÞ Lj < 12 L. Therefore (why?) it follows that if x 2 A \ V d ðcÞ, x 6¼ c, then f ðxÞ > 12 L > 0. If L < 0, a similar argument applies. Q.E.D. C04 12/09/2010 14:49:13 116 Page 116 CHAPTER 4 LIMITS Exercises for Section 4.2 1. Apply Theorem 4.2.4 to determine the following limits: x2 þ 2 lim ðx þ 1Þð2x þ 3Þ ðx 2 R Þ; (a) x!1 (b) lim ðx > 0Þ; x!1 x2 2 1 1 xþ1 ðx > 0Þ ; (d) lim 2 (c) lim ðx 2 R Þ: x!2 x þ 1 x!0 x þ 2 2x 2. Determine the following limits and state which theorems are used in each case. (You may wish to use Exercise 15 below.) rffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 4 2x þ 1 (b) lim (a) lim ðx > 0Þ; ðx > 0Þ; x!2 x 2 x!2 xþ3 p ffiffiffi 2 x1 (d) lim (c) lim ðx þ 1Þ 1 ðx > 0Þ; ðx > 0Þ: x!1 x 1 x!0 x pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 2x 1 þ 3x 3. Find lim where x > 0. x!0 x þ 2x2 4. Prove that lim cosð1=xÞ does not exist but that lim x cosð1=xÞ ¼ 0. x!0 x!0 5. Let f, g be defined on A R to R, and let c be a cluster point of A. Suppose that f is bounded on a neighborhood of c and that lim g ¼ 0. Prove that lim f g ¼ 0. x!c x!c 6. Use the definition of the limit to prove the first assertion in Theorem 4.2.4(a). 7. Use the sequential formulation of the limit to prove Theorem 4.2.4(b). 8. Let n 2 N be such that n 3. Derive the inequality x2 xn x2 for 1 < x < 1. Then use the fact that lim x2 ¼ 0 to show that lim xn ¼ 0. x!0 x!0 9. Let f, g be defined on A to R and let c be a cluster point of A. (a) Show that if both lim f and lim ð f þ gÞ exist, then lim g exists. x!c x!c x!c (b) If lim f and lim f g exist, does it follow that lim g exists? x!c x!c x!c 10. Give examples of functions f and g such that f and g do not have limits at a point c, but such that both f þ g and fg have limits at c. 11. Determine whether the following limits exist in R. lim sin 1=x2 ðx 6¼ 0Þ, lim x sin 1=x2 ðx 6¼ 0Þ, (a) x!0 (b) x!0 pffiffiffi 2 lim sgn sinð1=xÞ ðx 6¼ 0Þ, ðx > 0Þ. lim x sin 1=x (c) x!0 (d) x!0 12. Let f : R ! R be such that f ðx þ yÞ ¼ f ðxÞ þ f ðyÞ for all x, y in R. Assume that lim f ¼ L x!0 exists. Prove that L ¼ 0, and then prove that f has a limit at every point c 2 R. [Hint: First note that f ð2xÞ ¼ f ðxÞ þ f ðxÞ ¼ 2f ðxÞ for x 2 R. Also note that f ðxÞ ¼ f ðx cÞ þ f ðcÞ for x, c in R.] 13. Functions f and g are defined on R by f (x) :¼ x þ 1 and g (x) :¼ 2 if x 6¼ 1 and g(1) :¼ 0. (a) Find lim g (f (x)) and compare with the value of g (lim f (x)). x!1 x!1 (b) Find lim f (g (x)) and compare with the value of f (lim g (x)). x!1 x!1 14. Let A R, let f : A ! R and let c 2 R be a cluster point of A. If lim f exists, and if j f j denotes x!c the function defined for x 2 A by jf jðxÞ :¼ j f ðxÞj, prove that lim jf j ¼ lim f . x!c x!c 15. Let A R, let f : A ! R, andplet suppose pffiffiffiffiffiffiffiffithat ffi ffiffiffi c 2 R be a cluster point of A. In addition, pffiffiffi f ðxÞ 0 for all x 2 A, and p letffiffiffi f q beffiffiffiffiffiffiffiffiffiffi the function defined for x 2 A by ð f Þ (x) :¼ f ðxÞ. If lim f exists, prove that lim f ¼ lim f . x!c x!c x!c Section 4.3 Some Extensions of the Limit Concepty In this section, we shall present three types of extensions of the notion of a limit of a function that often occur. Since all the ideas here are closely parallel to ones we have already encountered, this section can be read easily. y This section can be largely omitted on a first reading of this chapter. C04 12/09/2010 14:49:13 Page 117 4.3 SOME EXTENSIONS OF THE LIMIT CONCEPT 117 One-Sided Limits There are times when a function f may not possess a limit at a point c, yet a limit does exist when the function is restricted to an interval on one side of the cluster point c. For example, the signum function considered in Example 4.1.10(b), and illustrated in Figure 4.1.2, has no limit at c ¼ 0. However, if we restrict the signum function to the interval (0, 1), the resulting function has a limit of 1 at c ¼ 0. Similarly, if we restrict the signum function to the interval (1, 0), the resulting function has a limit of 1 at c ¼ 0. These are elementary examples of right-hand and left-hand limits at c ¼ 0. 4.3.1 Definition Let A 2 R and let f : A ! R. (i) If c 2 R is a cluster point of the set A \ ðc; 1Þ ¼ fx 2 A: x > cg, then we say that L 2 R is a right-hand limit of f at c and we write lim f ¼ L x!cþ or lim f ðxÞ ¼ L x!cþ if given any e > 0 there exists a d ¼ dðeÞ > 0 such that for all x 2 A with 0 < x c < d, then j f ðxÞ Lj < e. (ii) If c 2 R is a cluster point of the set A \ ð1; cÞ ¼ fx 2 A: x < cg, then we say that L 2 R is a left-hand limit of f at c and we write lim f ¼ L x!c or lim f ðxÞ ¼ L x!c if given any e > 0 there exists a d > 0 such that for all x 2 A with 0 < c x < d, then j f ðxÞ Lj < e. Notes (1) The limits lim f and lim f are called one-sided limits of f at c. It is possible x!cþ x!c that neither one-sided limit may exist. Also, one of them may exist without the other existing. Similarly, as is the case for f ðxÞ :¼ sgnðxÞ at c ¼ 0, they may both exist and be different. (2) If A is an interval with left endpoint c, then it is readily seen that f : A ! R has a limit at c if and only if it has a right-hand limit at c. Moreover, in this case the limit lim f and the x!c right-hand limit lim f are equal. (A similar situation occurs for the left-hand limit when A x!cþ is an interval with right endpoint c.) The reader can show that f can have only one right-hand (respectively, left-hand) limit at a point. There are results analogous to those established in Sections 4.1 and 4.2 for twosided limits. In particular, the existence of one-sided limits can be reduced to sequential considerations. 4.3.2 Theorem Let A R, let f : A ! R, and let c 2 R be a cluster point of A \ ðc; 1Þ. Then the following statements are equivalent: (i) (ii) lim f ¼ L. x!cþ For every sequence (xn) that converges to c such that xn 2 A and xn > c for all n 2 N, the sequence ð f ðxn ÞÞ converges to L. C04 12/09/2010 14:49:13 118 Page 118 CHAPTER 4 LIMITS Figure4.3.2 Graph of hðxÞ ¼ 1= e1=x þ 1 ðx 6¼ 0Þ Figure 4.3.1 Graph of gðxÞ ¼ e1=x ðx 6¼ 0Þ We leave the proof of this result (and the formulation and proof of the analogous result for left-hand limits) to the reader. We will not take the space to write out the formulations of the one-sided version of the other results in Sections 4.1 and 4.2. The following result relates the notion of the limit of a function to one-sided limits. We leave its proof as an exercise. 4.3.3 Theorem Let A R, let f : A ! R, and let c 2 R be a cluster point of both of the sets A \ ðc; 1Þ and A \ ð1; cÞ. Then lim f ¼ L if and only if x!c lim f ¼ L ¼ lim f . x!cþ x!c 4.3.4 Examples (a) Let f ðxÞ :¼ sgnðxÞ. We have seen in Example 4.1.10(b) that sgn does not have a limit at 0. It is clear that lim sgnðxÞ ¼ þ1 and that lim sgnðxÞ ¼ 1. Since these one-sided limits are different, it x!0þ x!0 also follows from Theorem 4.3.3 that sgn(x) does not have a limit at 0. (b) Let gðxÞ :¼ e1=x for x 6¼ 0. (See Figure 4.3.1.) We first show that g does not have a finite right-hand limit at c ¼ 0 since it is not bounded on any right-hand neighborhood ð0; dÞ of 0. We shall make use of the inequality 0 < t < et ð1Þ for t > 0; which will be proved later (see Corollary 8.3.3). It follows from (1) that if x > 0, then 0 < 1=x < e1=x . Hence, if we take xn ¼ 1=n, then gðxn Þ > n for all n 2 N. Therefore lim e1=x does not exist in R. x!0þ However, lim e1=x ¼ 0. Indeed, if x < 0 and we take t ¼ 1=x in (1) we obtain x!0 0 < 1=x < e1=x . Since x < 0, this implies that 0 < e1=x < x for all x < 0. It follows from this inequality that lim e1=x ¼ 0. x!0 (c) Let hðxÞ :¼ 1= e1=x þ 1 for x 6¼ 0. (See Figure 4.3.2.) We have seen in part (b) that 0 < 1=x < e1=x for x > 0, whence 0< which implies that lim h ¼ 0. x!0þ 1 1 < x; < e1=x þ 1 e1=x C04 12/09/2010 14:49:14 Page 119 4.3 SOME EXTENSIONS OF THE LIMIT CONCEPT 119 Since we have seen in part (b) that lim e1=x ¼ 0, it follows from the analogue of x!0 Theorem 4.2.4(b) for left-hand limits that 1 1 1 lim ¼ 1: ¼ ¼ 1=x x!0 e1=x þ 1 0 þ 1 lim e þ 1 x!0 Note that for this function, both one-sided limits exist in R, but they are unequal. & Infinite Limits The function f ðxÞ :¼ 1=x2 for x 6¼ 0 (see Figure 4.3.3) is not bounded on a neighborhood of 0, so it cannot have a limit in the sense of Definition 4.1.4. While the symbols 1ð¼ þ1Þ and 1 do not represent real numbers, it is sometimes useful to be able to say that ‘‘f ðxÞ ¼ 1=x2 tends to 1 as x ! 0.’’ This use of 1 will not cause any difficulties, provided we exercise caution and never interpret 1 or 1 as being real numbers. Figure 4.3.3 f ðxÞ ¼ 1=x2 Graph of ðx 6¼ 0Þ Figure 4.3.4 Graph of gðxÞ ¼ 1=x ðx 6¼ 0Þ 4.3.5 Definition Let A R, let f : A ! R, and let c 2 R be a cluster point of A. (i) We say that f tends to 1 as x ! c, and write lim f ¼ 1; x!c (ii) if for every a 2 R there exists d ¼ dðaÞ > 0 such that for all x 2 A with 0 < jx cj < d, then f ðxÞ > a. We say that f tends to 1 as x ! c, and write lim f ¼ 1; x!c if for every b 2 R there exists d ¼ dðbÞ > 0 such that for all x 2 A with 0 < jx cj < d, then f ðxÞ < b. 4.3.6 Examples (a) lim 1=x2 ¼ 1. x!0 pffiffiffi For, if a > 0 is given, let d :¼ 1= a. It follows that if 0 < jxj < d, then x2 < 1=a so that 1=x2 > a. (b) Let gðxÞ :¼ 1=x for x 6¼ 0. (See Figure 4.3.4.) C04 12/09/2010 14:49:15 120 Page 120 CHAPTER 4 LIMITS The function g does not tend to either 1 or 1 as x ! 0. For, if a > 0 then gðxÞ < a for all x < 0, so that g does not tend to 1 as x ! 0. Similarly, if b < 0 then gðxÞ > b for all & x > 0, so that g does not tend to 1 as x ! 0. While many of the results in Sections 4.1 and 4.2 have extensions to this limiting notion, not all of them do since 1 are not real numbers. The following result is an analogue of the Squeeze Theorem 4.2.7. (See also Theorem 3.6.4.) 4.3.7 Theorem Let A R, let f ; g : A ! R, and let c 2 R be a cluster point of A. Suppose that f ðxÞ gðxÞ for all x 2 A; x 6¼ c. (a) If lim f ¼ 1, then lim g ¼ 1. x!c x!c (b) If lim g ¼ 1, then lim f ¼ 1. x!c x!c Proof. (a) If lim f ¼ 1 and a 2 R is given, then there exists dðaÞ > 0 such that if x!c 0 < jx cj < dðaÞ and x 2 A, then f ðxÞ > a. But since f ðxÞ gðxÞ for all x 2 A; x 6¼ c, it follows that if 0 < jx cj < dðaÞ and x 2 A, then gðxÞ > a. Therefore lim g ¼ 1. x!c The proof of (b) is similar. Q.E.D. The function gðxÞ ¼ 1=x considered in Example 4.3.6(b) suggests that it might be useful to consider one-sided infinite limits. We will define only right-hand infinite limits. 4.3.8 Definition Let A R and let f : A ! R. If c 2 R is a cluster point of the set A \ ðc; 1Þ ¼ fx 2 A : x > cg, then we say that f tends to 1 [respectively, 1] as x ! cþ, and we write h i lim f ¼ 1 respectively; lim f ¼ 1 ; x!cþ x!cþ if for every a 2 R there is d ¼ dðaÞ > 0 such that for all x 2 A with 0 < x c < d, then f ðxÞ > a [respectively, f ðxÞ < a]. 4.3.9 Examples (a) Let gðxÞ :¼ 1=x for x 6¼ 0. We have noted in Example 4.3.6(b) that lim g does not exist. However, it is an easy exercise to show that x!0 lim ð1=xÞ ¼ 1 and x!0þ lim ð1=xÞ ¼ 1: x!0 (b) It was seen in Example 4.3.4(b) that the function gðxÞ :¼ e1=x for x 6¼ 0 is not bounded on any interval ð0; dÞ; d > 0. Hence the right-hand limit of e1=x as x ! 0þ does not exist in the sense of Definition 4.3.1(i). However, since 1=x < e1=x it is readily seen that lim e x!0þ 1=x for x > 0; ¼ 1 in the sense of Definition 4.3.8. & Limits at Infinity It is also desirable to define the notion of the limit of a function as x ! 1. The definition as x ! 1 is similar. C04 12/09/2010 14:49:16 Page 121 4.3 SOME EXTENSIONS OF THE LIMIT CONCEPT 121 4.3.10 Definition Let A R and let f : A ! R. Suppose that ða; 1Þ A for some a 2 R. We say that L 2 R is a limit of f as x ! 1, and write lim f ¼ L or x!1 lim f ðxÞ ¼ L ; x!1 if given any e > 0 there exists K ¼ KðeÞ > a such that for any x > K, then j f ðxÞ Lj < e. The reader should note the close resemblance between 4.3.10 and the definition of a limit of a sequence. We leave it to the reader to show that the limits of f as x ! 1 are unique whenever they exist. We also have sequential criteria for these limits; we shall only state the criterion as x ! 1. This uses the notion of the limit of a properly divergent sequence (see Definition 3.6.1). 4.3.11 Theorem Let A R, let f : A ! R, and suppose that ða; 1Þ A for some a 2 R. Then the following statements are equivalent: (i) L ¼ lim f . x!1 (ii) For every sequence (xn) in A \ ða; 1Þ such that limðxn Þ ¼ 1, the sequence ð f ðxn ÞÞ converges to L. We leave it to the reader to prove this theorem and to formulate and prove the companion result concerning the limit as x ! 1. 4.3.12 Examples (a) Let gðxÞ :¼ 1=x for x 6¼ 0. It is an elementary exercise to show that lim ð1=xÞ ¼ 0 ¼ lim ð1=xÞ. (See Figure x!1 x!1 4.3.4.) (b) Let f ðxÞ :¼ 1=x2 for x 6¼ 0. The reader may show that lim ð1=x2 Þ ¼ 0 ¼ lim ð1=x2 Þ. (See Figure 4.3.3.) One x!1 x!1 way to do this is to show that if x 1 then 0 1=x2 1=x. In view of part (a), this implies that lim ð1=x2 Þ ¼ 0. & x!1 Just as it is convenient to be able to say that f ðxÞ ! 1 as x ! c for c 2 R, it is convenient to have the corresponding notion as x ! 1. We will treat the case where x ! 1. 4.3.13 Definition Let A R and let f : A ! R. Suppose that ða; 1Þ A for some a 2 A. We say that f tends to 1 [respectively, 1] as x ! 1, and write h i lim f ¼ 1 respectively; lim f ¼ 1 x!1 x!1 if given any a 2 R there exists K ¼ KðaÞ > a such that for any x > K, then f ðxÞ > a [respectively, f ðxÞ < a]. As before there is a sequential criterion for this limit. 4.3.14 Theorem Let A 2 R, let f : A ! R, and suppose that ða; 1Þ A for some a 2 R. Then the following statements are equivalent: C04 12/09/2010 14:49:16 122 (i) Page 122 CHAPTER 4 LIMITS lim f ¼ 1 ½respectively; lim f ¼ 1. x!1 x!1 (ii) For every sequence (xn) in ða; 1Þsuch that limðxn Þ ¼ 1, then limð f ðxn ÞÞ ¼ 1 ½respectively; limð f ðxn ÞÞ ¼ 1. The next result is an analogue of Theorem 3.6.5. 4.3.15 Theorem Let A R, let f ; g : A ! R, and suppose that ða; 1Þ A for some a 2 R. Suppose further that gðxÞ > 0 for all x > a and that for some L 2 R; L 6¼ 0, we have lim f ðxÞ x!1 gðxÞ ¼ L: (i) If L > 0, then lim f ¼ 1 if and only if lim g ¼ 1. (ii) If L < 0, then lim f ¼ 1 if and only if lim g ¼ 1. x!1 x!1 x!1 Proof. (i) x!1 Since L > 0, the hypothesis implies that there exists a1 > a such that 1 f ðxÞ 3 < L for x > a1 : 0< L 2 gðxÞ 2 Therefore we have 12 L gðxÞ < f ðxÞ < 32 L gðxÞ for all x > a1, from which the conclusion follows readily. The proof of (ii) is similar. Q.E.D. We leave it to the reader to formulate the analogous result as x ! 1. 4.3.16 Examples (a) lim xn ¼ 1 for n 2 N. x!1 Let gðxÞ :¼ xn for x 2 ð0; 1Þ. Given a 2 R, let K :¼ supf1; ag. Then for all x > K, we have gðxÞ ¼ xn x > a. Since a 2 R is arbitrary, it follows that lim g ¼ 1. x!1 lim xn ¼ 1 for n 2 N, n even, and lim xn ¼ 1 for n 2 N, n odd. x!1 x!1 We will treat the case n odd, say n ¼ 2k þ 1 with k ¼ 0, 1, . . . . Given a 2 R, let K :¼ inffa; 1g. For any x < K, then since ðx2 Þk 1, we have xn ¼ ðx2 Þk x x < a. Since a 2 R is arbitrary, it follows that lim xn ¼ 1. (b) x!1 (c) Let p : R ! R be the polynomial function pðxÞ :¼ an xn þ an1 xn1 þ þ a1 x þ a0 : Then lim p ¼ 1 if an > 0, and lim p ¼ 1 if an < 0. x!1 x!1 Indeed, let gðxÞ :¼ xn and apply Theorem 4.3.15. Since pðxÞ 1 1 ¼ an þ an1 þ þ a1 n1 gðxÞ x x þ a0 1 ; xn it follows that lim ðpðxÞ=gðxÞÞ ¼ an . Since lim g ¼ 1, the assertion follows from x!1 x!1 Theorem 4.3.15. (d) Let p be the polynomial function in part (c). Then lim p ¼ 1 [respectively, 1] if x!1 n is even [respectively, odd] and an > 0. & We leave the details to the reader. C04 12/09/2010 14:49:17 Page 123 4.3 SOME EXTENSIONS OF THE LIMIT CONCEPT 123 Exercises for Section 4.3 1. Prove Theorem 4.3.2. 2. Give an example of a function that has a right-hand limit but not a left-hand limit at a point. 3. Let f ðxÞ :¼ jxj1=2 for x 6¼ 0. Show that lim f ðxÞ ¼ lim f ðxÞ ¼ þ1. x!0þ x!0 4. Let c 2 R and let f be defined for x 2 ðc; 1Þ and f ðxÞ > 0 for all x 2 ðc; 1Þ. Show that lim f ¼ 1 if and only if lim 1=f ¼ 0. x!c x!c 5. Evaluate the following limits, or show that they do not exist. x x ðx 6¼ 1Þ, ðx 6¼ 1Þ, (a) lim (b) lim x!1þ x 1 pffiffiffi x!1 x 1 pffiffiffi lim ðx þ 2Þ= x ðx > 0Þ, lim ðx þ 2Þ= x ðx > 0Þ, (c) x!0þ (d) x!1 pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi lim x þ 1 =x ðx > 1Þ, lim x þ 1 =x ðx > 0Þ, (f) x!1 (e) x!0 pffiffiffi pffiffiffi x5 xx (g) lim pffiffiffi (h) lim pffiffiffi ðx > 0Þ, ðx > 0Þ. x!1 x þ 3 x!1 x þ x 6. Prove Theorem 4.3.11. 7. Suppose that f and g have limits in R as x ! 1 and that f ðxÞ gðxÞ for all x 2 ða; 1Þ. Prove that lim f lim g. x!1 x!1 8. Let f be defined on (0, 1) to R. Prove that lim f ðxÞ ¼ L if and only if lim f ð1=xÞ ¼ L. x!1 x!0þ 9. Show that if f : ða; 1Þ ! R is such that lim xf ðxÞ ¼ L where L 2 R, then lim f ðxÞ ¼ 0. x!1 x!1 10. Prove Theorem 4.3.14. 11. Suppose that lim f ðxÞ ¼ L where L > 0, and that lim gðxÞ ¼ 1. Show that lim f ðxÞgðxÞ ¼ 1. x!c x!c If L ¼ 0, show by example that this conclusion may fail. x!c 12. Find functions f and g defined on (0, 1) such that lim f ¼ 1 and lim g ¼ 1, and x!1 x!1 lim ð f gÞ ¼ 0. Can you find such functions, with g(x) > 0 for all x 2 ð0; 1Þ, such that x!1 lim f =g ¼ 0? x!1 13. Let f and g be defined on (a, 1) and suppose lim f ¼ L and lim g ¼ 1. Prove that x!1 x!1 lim f g ¼ L. x!1 C05 12/08/2010 14:19:37 Page 124 CHAPTER 5 CONTINUOUS FUNCTIONS We now begin the study of the most important class of functions that arises in real analysis: the class of continuous functions. The term ‘‘continuous’’ has been used since the time of Newton to refer to the motion of bodies or to describe an unbroken curve, but it was not made precise until the nineteenth century. Work of Bernhard Bolzano in 1817 and Augustin-Louis Cauchy in 1821 identified continuity as a very significant property of functions and proposed definitions, but since the concept is tied to that of limit, it was the careful work of Karl Weierstrass in the 1870s that brought proper understanding to the idea of continuity. We will first define the notions of continuity at a point and continuity on a set, and then show that various combinations of continuous functions give rise to continuous functions. Then in Section 5.3 we establish the fundamental properties that make continuous functions so important. For instance, we will prove that a continuous function on a closed bounded interval must attain a maximum and a minimum value. We also prove that a continuous function must take on every value intermediate to any two values it attains. These properties and others are not possessed by general functions, as various examples illustrate, and thus they distinguish continuous functions as a very special class of functions. In Section 5.4 we introduce the very important notion of uniform continuity. The distinction between continuity and uniform continuity is somewhat subtle and was not fully appreciated until the work of Weierstrass and the mathematicians of his era, but it proved to be very significant in applications. We present one application to the idea of approximating continuous functions by more elementary functions (such as polynomials). Karl Weierstrass Karl Weierstrass (¼Weierstrab) (1815–1897) was born in Westphalia, Germany. His father, a customs officer in a salt works, insisted that he study law and public finance at the University of Bonn, but he had more interest in drinking and fencing, and left Bonn without receiving a diploma. He then enrolled in the Academy of M€unster where he studied mathematics with Christoph Gudermann. From 1841 to 1854 he taught at various gymnasia in Prussia. Despite the fact that he had no contact with the mathematical world during this time, he worked hard on mathematical research and was able to publish a few papers, one of which attracted considerable attention. Indeed, the University of K€onigsberg gave him an honorary doctoral degree for this work in 1855. The next year, he secured positions at the Industrial Institute of Berlin and the University of Berlin. He remained at Berlin until his death. A methodical and painstaking scholar, Weierstrass distrusted intuition and worked to put everything on a firm and logical foundation. He did fundamental work on the foundations of arithmetic and analysis, on complex analysis, the calculus of variations, and algebraic geometry. Due to his meticulous preparation, he was an extremely popular lecturer; it was not unusual for him to speak about advanced mathematical topics to audiences of more than 250. Among his auditors are counted Georg Cantor, Sonya Kovalevsky, G€ osta Mittag-Leffler, Max Planck, Otto H€older, David Hilbert, and Oskar Bolza (who had many American doctoral students). Through his writings and his lectures, Weierstrass had a profound influence on contemporary mathematics. 124 C05 12/08/2010 14:19:38 Page 125 5.1 CONTINUOUS FUNCTIONS 125 The notion of a ‘‘gauge’’ is introduced in Section 5.5 and is used to provide an alternative method of proving the fundamental properties of continuous functions. The main significance of this concept, however, is in the area of integration theory where gauges are essential in defining the generalized Riemann integral. This will be discussed in Chapter 10. Monotone functions are an important class of functions with strong continuity properties and they are discussed in Section 5.6. Section 5.1 Continuous Functions In this section, which is very similar to Section 4.1, we will define what it means to say that a function is continuous at a point, or on a set. This notion of continuity is one of the central concepts of mathematical analysis, and it will be used in almost all of the following material in this book. Consequently, it is essential that the reader master it. 5.1.1 Definition Let A R, let f : A ! R, and let c 2 A. We say that f is continuous at c if, given any number e > 0, there exists d > 0 such that if x is any point of A satisfying jx cj < d, then j f ðxÞ f ðcÞj < e. If f fails to be continuous at c, then we say that f is discontinuous at c. As with the definition of limit, the definition of continuity at a point can be formulated very nicely in terms of neighborhoods. This is done in the next result. We leave the verification as an important exercise for the reader. See Figure 5.1.1. Figure 5.1.1 Given V e ð f ðcÞÞ, a neighborhood V d ðcÞ is to be determined 5.1.2 Theorem A function f : A ! R is continuous at a point c 2 A if and only if given any e-neighborhood V e ð f ðcÞÞ of f (c) there exists a d-neighborhood V d ðcÞ of c such that if x is any point of A \ V d ðcÞ, then f (x) belongs to V e ð f ðcÞÞ, that is, f ðA \ V d ðcÞÞ V e ð f ðcÞÞ: Remarks (1) If c 2 A is a cluster point of A, then a comparison of Definitions 4.1.4 and 5.1.1 show that f is continuous at c if and only if ð1Þ f ðcÞ ¼ lim f ðxÞ: x!c C05 12/08/2010 14:19:39 126 Page 126 CHAPTER 5 CONTINUOUS FUNCTIONS Thus, if c is a cluster point of A, then three conditions must hold for f to be continuous at c: (i) f must be defined at c (so that f (c) makes sense), (ii) the limit of f at c must exist in R (so that lim f ðxÞ makes sense), and x!c (iii) these two values must be equal. (2) If c 2 A is not a cluster point of A, then there exists a neighborhood V d ðcÞ of c such that A \ V d ðcÞ ¼ fcg. Thus we conclude that a function f is automatically continuous at a point c 2 A that is not a cluster point of A. Such points are often called ‘‘isolated points’’ of A. They are of little practical interest to us, since they have no relation to a limiting process. Since continuity is automatic for such points, we generally test for continuity only at cluster points. Thus we regard condition (1) as being characteristic for continuity at c. A slight modification of the proof of Theorem 4.1.8 for limits yields the following sequential version of continuity at a point. 5.1.3 Sequential Criterion for Continuity A function f : A ! R is continuous at the point c 2 A if and only if for every sequence (xn) in A that converges to c, the sequence ð f ðxn ÞÞ converges to f (c). The following Discontinuity Criterion is a consequence of the last theorem. It should be compared with the Divergence Criterion 4.1.9(a) with L ¼ f (c). Its proof should be written out in detail by the reader. 5.1.4 Discontinuity Criterion Let A R, let f : A ! R, and let c 2 A. Then f is discontinuous at c if and only if there exists a sequence (xn) in A such that (xn) converges to c, but the sequence ð f ðxn ÞÞ does not converge to f (c). So far we have discussed continuity at a point. To talk about the continuity of a function on a set, we will simply require that the function be continuous at each point of the set. We state this formally in the next definition. 5.1.5 Definition Let A R and let f : A ! R. If B is a subset of A, we say that f is continuous on the set B if f is continuous at every point of B. 5.1.6 Examples (a) The constant function f ðxÞ :¼ b is continuous on R. It was seen in Example 4.1.7(a) that if c 2 R, then lim f ðxÞ ¼ b. Since f ðcÞ ¼ b, we x!c have lim f ðxÞ ¼ f ðcÞ, and thus f is continuous at every point c 2 R. Therefore f is x!c continuous on R. (b) gðxÞ :¼ x is continuous on R. It was seen in Example 4.1.7(b) that if c 2 R, then we have lim g ¼ c. Since gðcÞ ¼ c, x!c then g is continuous at every point c 2 R. Thus g is continuous on R. (c) hðxÞ :¼ x2 is continuous on R. It was seen in Example 4.1.7(c) that if c 2 R, then we have lim h ¼ c2 . Since x!c hðcÞ ¼ c2 , then h is continuous at every point c 2 R. Thus h is continuous on R. (d) wðxÞ :¼ 1=x is continuous on A :¼ fx 2 R : x > 0g. It was seen in Example 4.1.7(d) that if c 2 A, then we have lim w ¼ 1=c. Since x!c wðcÞ ¼ 1=c, this shows that w is continuous at every point c 2 A. Thus w is continuous on A. C05 12/08/2010 14:19:39 Page 127 5.1 CONTINUOUS FUNCTIONS 127 (e) wðxÞ :¼ 1=x is not continuous at x ¼ 0. Indeed, if wðxÞ ¼ 1=x for x > 0, then w is not defined for x ¼ 0, so it cannot be continuous there. Alternatively, it was seen in Example 4.1.10(a) that lim w does not exist x!0 in R, so w cannot be continuous at x ¼ 0. (f) The signum function sgn is not continuous at 0. The signum function was defined in Example 4.1.10(b), where it was also shown that lim sgnðxÞ does not exist in R. Therefore sgn is not continuous at x ¼ 0 (even though sgn 0 x!0 is defined). It is an exercise to show that sgn is continuous at every point c 6¼ 0. Note In the next two examples, we introduce functions that played a significant role in the development of real analysis. Discontinuities are emphasized and it is not possible to graph either of them satisfactorily. The intuitive idea of drawing a curve in the plane to represent a function simply does not apply, and plotting a handful of points gives only a hint of their character. In the nineteenth century, these functions clearly demonstrated the need for a precise and rigorous treatment of the basic concepts of analysis. They will reappear in later sections. (g) Let A :¼ R and let f be Dirichlet’s ‘‘discontinuous function’’ defined by f ðxÞ :¼ 1 if 0 if x is rational; x is irrational: We claim that f is not continuous at any point of R. (This function was introduced in 1829 by P. G. L. Dirichlet.) Indeed, if c is a rational number, let (xn) be a sequence of irrational numbers that converges to c. (Corollary 2.4.9 to the Density Theorem 2.4.8 assures us that such a sequence does exist.) Since f ðxn Þ ¼ 0 for all n 2 N, we have limð f ðxn ÞÞ ¼ 0, while f ðcÞ ¼ 1. Therefore f is not continuous at the rational number c. On the other hand, if b is an irrational number, let ( yn) be a sequence of rational numbers that converge to b. (The Density Theorem 2.4.8 assures us that such a sequence does exist.) Since f ðyn Þ ¼ 1 for all n 2 N, we have limð f ð yn ÞÞ ¼ 1, while f ðbÞ ¼ 0. Therefore f is not continuous at the irrational number b. Since every real number is either rational or irrational, we deduce that f is not continuous at any point in R. Figure 5.1.2 Thomae’s function C05 12/08/2010 14:19:40 128 Page 128 CHAPTER 5 CONTINUOUS FUNCTIONS (h) Let A :¼ fx 2 R : x > 0g. For any irrational number x > 0 we define hðxÞ :¼ 0. For a rational number in A of the form m=n, with natural numbers m, n having no common factors except 1, we define hðm=nÞ :¼ 1=n. (We also define hð0Þ :¼ 1.) We claim that h is continuous at every irrational number in A, and is discontinuous at every rational number in A. (This function was introduced in 1875 by K. J. Thomae.) Indeed, if a > 0 is rational, let (xn) be a sequence of irrational numbers in A that converges to a. Then limðhðxn ÞÞ ¼ 0, while h(a) > 0. Hence h is discontinuous at a. On the other hand, if b is an irrational number and e > 0, then (by the Archimedean Property) there is a natural number n0 such that 1=n0 < e. There are only a finite number of rationals with denominator less than n0 in the interval ðb 1; b þ 1Þ. (Why?) Hence d > 0 can be chosen so small that the neighborhood ðb d; b þ dÞ contains no rational numbers with denominator less than n0. It then follows that for jx bj < d; x 2 A, we have jhðxÞ hðbÞj ¼ jhðxÞj 1=n0 < e. Thus h is continuous at the irrational number b. Consequently, we deduce that Thomae’s function h is continuous precisely at the & irrational points in A. (See Figure 5.1.2.) 5.1.7 Remarks (a) Sometimes a function f : A ! R is not continuous at a point c because it is not defined at this point. However, if the function f has a limit L at the point c and if we define F on A [ fcg ! R by FðxÞ :¼ L f ðxÞ for for x ¼ c; x 2 A; then F is continuous at c. To see this, one needs to check that lim F ¼ L, but this follows x!c (why?), since lim f ¼ L. x!c (b) If a function g : A ! R does not have a limit at c, then there is no way that we can obtain a function G : A [ fcg ! R that is continuous at c by defining GðxÞ :¼ C gðxÞ for for x ¼ c; x 2 A: To see this, observe that if lim G exists and equals C, then lim g must also exist and x!c x!c equal C. 5.1.8 Examples (a) The function gðxÞ :¼ sinð1=xÞ for x 6¼ 0 (see Figure 4.1.3) does not have a limit at x ¼ 0 (see Example 4.1.10(c)). Thus there is no value that we can assign at x ¼ 0 to obtain a continuous extension of g at x ¼ 0. (b) Let f ðxÞ :¼ x sin ð1=xÞ for x 6¼ 0. (See Figure 5.1.3.) It was seen in Example 4.2.8(f) that lim ðx sinð1=xÞÞ ¼ 0. Therefore it follows from Remark 5.1.7(a) that if we define x!0 F : R ! R by FðxÞ :¼ then F is continuous at x ¼ 0. 0 for x sinð1=xÞ for x ¼ 0; x 6¼ 0; & C05 12/08/2010 14:19:41 Page 129 5.1 CONTINUOUS FUNCTIONS Figure 5.1.3 129 Graph of f ðxÞ ¼ x sinð1=xÞ ðx 6¼ 0Þ Exercises for Section 5.1 1. Prove the Sequential Criterion 5.1.3. 2. Establish the Discontinuity Criterion 5.1.4. 3. Let a < b < c. Suppose that f is continuous on [a, b], that g is continuous on [b, c], and that f ðbÞ ¼ gðbÞ. Define h on [a, c] by hðxÞ :¼ f ðxÞ for x 2 ½a; b and hðxÞ :¼ gðxÞ for x 2 ½b; c. Prove that h is continuous on [a, c]. 4. If x 2 R, we define vxb to be the greatest integer n 2 Z such that n x. (Thus, for example, v8:3b ¼ 8; vpb ¼ 3; v pb ¼ 4.) The function x 7! vxb is called the greatest integer function. Determine the points of continuity of the following functions: (a) f ðxÞ :¼ vxb; (b) gðxÞ :¼ x vxb; (c) hðxÞ :¼ vsin xb; (d) kðxÞ :¼ v1=xb ðx 6¼ 0Þ: 5. Let f be defined for all x 2 R; x 6¼ 2, by f ðxÞ ¼ ðx2 þ x 6Þ=ðx 2Þ. Can f be defined at x ¼ 2 in such a way that f is continuous at this point? 6. Let A R and let f : A ! R be continuous at a point c 2 A. Show that for any e > 0, there exists a neighborhood V d ðcÞ of c such that if x; y 2 A \ V d ðcÞ, then j f ðxÞ f ðyÞj < e. 7. Let f : R ! R be continuous at c and let f ðcÞ > 0. Show that there exists a neighborhood V d ðcÞ of c such that if x 2 V d ðcÞ, then f ðxÞ > 0. 8. Let f : R ! R be continuous on R and let S :¼ fx 2 R : f ðxÞ ¼ 0g be the ‘‘zero set’’ of f. If ðxn Þ is in S and x ¼ limðxn Þ, show that x 2 S. 9. Let A B R, let f : B ! R and let g be the restriction of f to A (that is, gðxÞ ¼ f ðxÞ for x 2 A). (a) If f is continuous at c 2 A, show that g is continuous at c. (b) Show by example that if g is continuous at c, it need not follow that f is continuous at c. 10. Show that the absolute value function f ðxÞ :¼ jxj is continuous at every point c 2 R. 11. Let K > 0 and let f : R ! R satisfy the condition j f ðxÞ f ðyÞj Kjx yj for all x; y 2 R. Show that f is continuous at every point c 2 R. 12. Suppose that f : R ! R is continuous on R and that f ðrÞ ¼ 0 for every rational number r. Prove that f ðxÞ ¼ 0 for all x 2 R. 13. Define g : R ! R by gðxÞ :¼ 2x for x rational, and gðxÞ :¼ x þ 3 for x irrational. Find all points at which g is continuous. C05 12/08/2010 14:19:42 130 Page 130 CHAPTER 5 CONTINUOUS FUNCTIONS 14. Let A :¼ ð0; 1Þ and let k : A ! R be defined as follows. For x 2 A; x irrational, we define kðxÞ ¼ 0; for x 2 A rational and of the form x ¼ m=n with natural numbers m, n having no common factors except 1, we define kðxÞ :¼ n. Prove that k is unbounded on every open interval in A. Conclude that k is not continuous at any point of A. (See Example 5.1.6(h).) 15. Let f : ð0; 1Þ ! R be bounded but such that lim f does not exist. Show that there are two x!0 sequences ðxn Þ and ðyn Þ in (0, 1) with limðxn Þ ¼ 0 ¼ limðyn Þ, but such that ð f ðxn ÞÞ and ð f ðyn ÞÞ exist but are not equal. Section 5.2 Combinations of Continuous Functions Let A R and let f and g be functions that are defined on A to R and let b 2 R. In Definition 4.2.3 we defined the sum, difference, product, and multiple functions denoted by f þ g; f g; f g; bf . In addition, if h : A ! R is such that hðxÞ 6¼ 0 for all x 2 A, then we defined the quotient function denoted by f =h. The next result is similar to Theorem 4.2.4, from which it follows. 5.2.1 Theorem Let A R, let f and g be functions on A to R, and let b 2 R. Suppose that c 2 A and that f and g are continuous at c. (a) Then f þ g; f g; f g, and bf are continuous at c. (b) If h : A ! R is continuous at c 2 A and if hðxÞ 6¼ 0 for all x 2 A, then the quotient f=h is continuous at c. Proof. If c 2 A is not a cluster point of A, then the conclusion is automatic. Hence we assume that c is a cluster point of A. (a) Since f and g are continuous at c, then f ðcÞ ¼ lim f and gðcÞ ¼ lim g: x!c x!c Hence it follows from Theorem 4.2.4(a) that ð f þ gÞðcÞ ¼ f ðcÞ þ gðcÞ ¼ lim ð f þ gÞ: x!c Therefore f þ g is continuous at c. The remaining assertions in part (a) are proved in a similar fashion. (b) Since c 2 A, then hðcÞ 6¼ 0. But since hðcÞ ¼ lim h, it follows from Theorem 4.2.4(b) x!c that lim f f f ðcÞ x!c f ðcÞ ¼ ¼ ¼ lim : h hðcÞ lim h x!c h x!c Therefore f=h is continuous at c. Q.E.D. The next result is an immediate consequence of Theorem 5.2.1, applied to every point of A. However, since it is an extremely important result, we shall state it formally. 5.2.2 Theorem Let A R, let f and g be continuous on A to R, and let b 2 R. (a) The functions f þ g; f g; f g, and bf are continuous on A. (b) If h : A ! R is continuous on A and hðxÞ 6¼ 0 for x 2 A, then the quotient f=h is continuous on A. C05 12/08/2010 14:19:43 Page 131 5.2 COMBINATIONS OF CONTINUOUS FUNCTIONS 131 Remark To define quotients, it is sometimes more convenient to proceed as follows. If w : A ! R, let A1 :¼ fx 2 A : wðxÞ 6¼ 0g. We can define the quotient f =w on the set A1 by ð1Þ f f ðxÞ ðxÞ :¼ w w ðxÞ for x 2 A1 : If w is continuous at a point c 2 A1 , it is clear that the restriction w1 of w to A1 is also continuous at c. Therefore it follows from Theorem 5.2.1(b) applied to w1 that f =w1 is continuous at c 2 A. Since ð f =wÞðxÞ ¼ ð f =w1 ÞðxÞ for x 2 A1 it follows that f =w is continuous at c 2 A1 . Similarly, if f and w are continuous on A, then the function f =w, defined on A1 by (1), is continuous on A1. 5.2.3 Examples (a) Polynomial functions. If p is a polynomial function, so that pðxÞ ¼ an xn þ an1 xn1 þ þ a1 x þ a0 for all x 2 R, then it follows from Example 4.2.5(f) that pðcÞ ¼ lim p for any c 2 R. Thus a x!c polynomial function is continuous on R. (b) Rational functions. If p and q are polynomial functions on R, then there are at most a finite number a1 ; . . . ; am of real roots of q. If x 2 = fa1 ; . . . ; am g then qðxÞ 6¼ 0 so that we can define the rational function r by rðxÞ :¼ pðxÞ qðxÞ for x2 = fa1 ; . . . ; am g : It was seen in Example 4.2.5(g) that if qðcÞ 6¼ 0, then rðcÞ ¼ pðcÞ pðxÞ ¼ lim ¼ lim rðxÞ: qðcÞ x!c qðxÞ x!c In other words, r is continuous at c. Since c is any real number that is not a root of q, we infer that a rational function is continuous at every real number for which it is defined. (c) We shall show that the sine function sin is continuous on R. To do so we make use of the following properties of the sine and cosine functions. (See Section 8.4.) For all x, y, z 2 R we have: jsin zj jzj; jcos zj 1; sin x sin y ¼ 2 sin 12ðx yÞ cos 12ðx þ yÞ : Hence if c 2 R, then we have jsin x sin cj 2 12 jx cj 1 ¼ jx cj: Therefore sin is continuous at c. Since c 2 R is arbitrary, it follows that sin is continuous on R. (d) The cosine function is continuous on R. We make use of the following properties of the sine and cosine functions. For all x; y; z 2 R we have: jsin zj jzj; jsin zj 1; cos x cos y ¼ 2 sin 12ðx þ yÞ sin 12ðx yÞ : Hence if c 2 R, then we have jcos x cos cj 2 1 12jc xj ¼ jx cj: C05 12/08/2010 14:19:43 132 Page 132 CHAPTER 5 CONTINUOUS FUNCTIONS Therefore cos is continuous at c. Since c 2 R is arbitrary, it follows that cos is continuous on R. (Alternatively, we could use the relation cos x ¼ sinðx þ p=2Þ.) (e) The functions tan, cot, sec, csc are continuous where they are defined. For example, the cotangent function is defined by cos x cot x :¼ sin x provided sin x 6¼ 0 (that is, provided x 6¼ np; n 2 Z). Since sin and cos are continuous on R, it follows (see the Remark before Example 5.2.3) that the function cot is continuous & on its domain. The other trigonometric functions are treated similarly. 5.2.4 Theorem Let A R, let f : A ! R, and let j f j be defined by j f jðxÞ :¼ j f ðxÞj for x 2 A. (a) If f is continuous at a point c 2 A, then j f j is continuous at c. (b) If f is continuous on A, then j f j is continuous on A. Proof. This is an immediate consequence of Exercise 4.2.14. 5.2.5 Theorem Let Ap A !ffi R, and let f ðxÞ 0 for all x 2 A. We let ffiffiffi R, let fp: ffiffiffiffiffiffiffiffi defined for x 2 A by ð f ÞðxÞ :¼ f ðxÞ. pffiffiffi (a) If f is continuous at a point c 2 A, then f is continuous at c. pffiffiffi (b) If f is continuous on A, then f is continuous on A. Proof. This is an immediate consequence of Exercise 4.2.15. Q.E.D. pffiffiffi f be Q.E.D. Composition of Continuous Functions We now show that if the function f : A ! R is continuous at a point c and if g : B ! R is continuous at b ¼ f (c), then the composition g f is continuous at c. In order to assure that g f is defined on all of A, we also need to assume that f ðAÞ B. 5.2.6 Theorem Let A, B R and let f : A ! R and g : B ! R be functions such that f ðAÞ B. If f is continuous at a point c 2 A and g is continuous at b ¼ f ðcÞ 2 B, then the composition g f : A ! R is continuous at c. Proof. Let W be an e-neighborhood of g(b). Since g is continuous at b, there is a d-neighborhood V of b ¼ f ðcÞ such that if y 2 B \ V then gðyÞ 2 W. Since f is continuous at c, there is a g-neighborhood U of c such that if x 2 A \ U, then f ðxÞ 2 V. (See Figure 5.2.1.) Since f ðAÞ B, it follows that if x 2 A \ U, then f ðxÞ 2 B \ V so that g f ðxÞ ¼ gð f ðxÞÞ 2 W. But since W is an arbitrary e-neighborhood of g(b), this implies that g f is continuous at c. Q.E.D. 5.2.7 Theorem Let A; B R, let f : A ! R be continuous on A, and let g : B ! R be continuous on B. If f ðAÞ B, then the composite function g f : A ! R is continuous on A. Proof. The theorem follows immediately from the preceding result, if f and g are continuous at every point of A and B, respectively. Q.E.D. C05 12/08/2010 14:19:44 Page 133 5.2 COMBINATIONS OF CONTINUOUS FUNCTIONS Figure 5.2.1 133 The composition of f and g Theorems 5.2.6 and 5.2.7 are very useful in establishing that certain functions are continuous. They can be used in many situations where it would be difficult to apply the definition of continuity directly. 5.2.8 Examples (a) Let g1 ðxÞ :¼ jxj for x 2 R. It follows from the Triangle Inequality that jg1 ðxÞ g1 ðcÞj jx cj for all x; c 2 R. Hence g1 is continuous at c 2 R. If f : A ! R is any function that is continuous on A, then Theorem 5.2.7 implies that g1 f ¼ j f j is continuous on A. This gives another proof of Theorem 5.2.4. pffiffiffi (b) Let g2 ðxÞ :¼ x for x 0. It follows from Theorems 3.2.10 and 5.1.3 that g2 is continuous at any number c 0. If f : A ! R is continuous pffiffiffi on A and if f ðxÞ 0 for all x 2 A, then it follows from Theorem 5.2.7 that g2 f ¼ f is continuous on A. This gives another proof of Theorem 5.2.5. (c) Let g3 ðxÞ :¼ sin x for x 2 R. We have seen in Example 5.2.3(c) that g3 is continuous on R. If f : A ! R is continuous on A, then it follows from Theorem 5.2.7 that g3 f is continuous on A. In particular, if f ðxÞ :¼ 1=x for x 6¼ 0, then the function gðxÞ :¼ sinð1=xÞ is continuous at every point c 6¼ 0. [We have seen, in Example 5.1.8(a), that g cannot be defined at & 0 in order to become continuous at that point.] Exercises for Section 5.2 1. Determine the points of continuity of the following functions and state which theorems are used in each case. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 þ 2x þ 1 pffiffiffiffi ðx 2 R Þ; (b) gðxÞ :¼ x þ x ðx 0Þ; (a) f ðxÞ :¼ x2 þ 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi (d) kðxÞ :¼ cos 1 þ x2 ðx 2 R Þ: (c) hðxÞ :¼ 1 þ j sin xj ðx 6¼ 0Þ; x 2. Show that if f : A ! R is continuous on A R and if n 2 N, then the function f n defined by f n ðxÞ ¼ ð f ðxÞÞn , for x 2 A, is continuous on A. 3. Give an example of functions f and g that are both discontinuous at a point c in R such that (a) the sum f þ g is continuous at c, (b) the product fg is continuous at c. C05 12/08/2010 14:19:45 134 Page 134 CHAPTER 5 CONTINUOUS FUNCTIONS 4. Let x 7! vxb denote the greatest integer function (see Exercise 5.1.4). Determine the points of continuity of the function f ðxÞ :¼ x vxb; x 2 R. 5. Let g be defined on R by gð1Þ :¼ 0, and gðxÞ :¼ 2 if x 6¼ 1, and let f ðxÞ :¼ x þ 1 for all x 2 R. Show that lim g f 6¼ ðg f Þð0Þ. Why doesn’t this contradict Theorem 5.2.6? x!0 6. Let f, g be defined on R and let c 2 R. Suppose that lim f ¼ b and that g is continuous at b. x!c Show that lim g f ¼ gðbÞ. (Compare this result with Theorem 5.2.7 and the preceding x!c exercise.) 7. Give an example of a function f : ½0; 1 ! R that is discontinuous at every point of [0, 1] but such that j f j is continuous on [0, 1]. 8. Let f, g be continuous from R to R, and suppose that f ðrÞ ¼ gðrÞ for all rational numbers r. Is it true that f ðxÞ ¼ gðxÞ for all x 2 R? 9. Let h : R ! R be continuous on R satisfying hðm=2n Þ ¼ 0 for all m 2 Z; n 2 N. Show that hðxÞ ¼ 0 for all x 2 R. 10. Let f : R ! R be continuous on R, and let P :¼ fx 2 R : f ðxÞ > 0g. If c 2 P, show that there exists a neighborhood V d ðcÞ P. 11. If f and g are continuous on R, let S :¼ fx 2 R : f ðxÞ gðxÞg. If ðsn Þ S and limðsn Þ ¼ s, show that s 2 S. 12. A function f : R ! R is said to be additive if f ðx þ yÞ ¼ f ðxÞ þ f ðyÞ for all x, y in R. Prove that if f is continuous at some point x0, then it is continuous at every point of R. (See Exercise 4.2.12.) 13. Suppose that f is a continuous additive function on R. If c :¼ f ð1Þ, show that we have f ðxÞ ¼ cx for all x 2 R. [Hint: First show that if r is a rational number, then f ðrÞ ¼ cr.] 14. Let g : R ! R satisfy the relation gðx þ yÞ ¼ gðxÞ gðyÞ for all x, y in R. Show that if g is continuous at x ¼ 0, then g is continuous at every point of R. Also if we have gðaÞ ¼ 0 for some a 2 R, then gðxÞ ¼ 0 for all x 2 R. 15. Let f ; g : R ! R be continuous at a point c, and let hðxÞ :¼ supf f ðxÞ; gðxÞg for x 2 R. Show that hðxÞ ¼ 12 ð f ðxÞ þ gðxÞÞ þ 12 j f ðxÞ gðxÞj for all x 2 R. Use this to show that h is continuous at c. Section 5.3 Continuous Functions on Intervals Functions that are continuous on intervals have a number of very important properties that are not possessed by general continuous functions. In this section, we will establish some deep results that are of considerable importance and that will be applied later. Alternative proofs of these results will be given in Section 5.5. 5.3.1 Definition A function f : A ! R is said to be bounded on A if there exists a constant M > 0 such that j f ðxÞj M for all x 2 A. In other words, a function is bounded on a set if its range is a bounded set in R. To say that a function is not bounded on a given set is to say that no particular number can serve as a bound for its range. In exact language, a function f is not bounded on the set A if given any M > 0, there exists a point xM 2 A such that j f ðxM Þj > M. We often say that f is unbounded on A in this case. For example, the function f defined on the interval A :¼ ð0; 1Þ by f ðxÞ :¼ 1=x is not bounded on A because for any M > 0 we can take the point xM :¼ 1=ðM þ 1Þ in A to get f ðxM Þ ¼ 1=xM ¼ M þ 1 > M. This example shows that continuous functions need not be C05 12/08/2010 14:19:45 Page 135 5.3 CONTINUOUS FUNCTIONS ON INTERVALS 135 bounded. In the next theorem, however, we show that continuous functions on a certain type of interval are necessarily bounded. 5.3.2 Boundedness Theoremy Let I :¼ ½a; b be a closed bounded interval and let f : I ! R be continuous on I. Then f is bounded on I. Proof. Suppose that f is not bounded on I. Then, for any n 2 N there is a number xn 2 I such that j f ðxn Þj > n. Since I is bounded, the sequence X :¼ ðxn Þ is bounded. Therefore, the Bolzano-Weierstrass Theorem 3.4.8 implies that there is a subsequence X 0 ¼ ðxnr Þ of X that converges to a number x. Since I is closed and the elements of X 0 belong to I, it follows from Theorem 3.2.6 that x 2 I. Then f is continuous at x, so that ð f ðxnr ÞÞ converges to f ðxÞ. We then conclude from Theorem 3.2.2 that the convergent sequence ðf ðxnr ÞÞ must be bounded. But this is a contradiction since j f ðxnr Þj > nr r for r 2 N: Therefore the supposition that the continuous function f is not bounded on the closed bounded interval I leads to a contradiction. Q.E.D. To show that each hypothesis of the Boundedness Theorem is needed, we can construct examples that show the conclusion fails if any one of the hypotheses is relaxed. (i) The interval must be bounded. The function f ðxÞ :¼ x for x in the unbounded, closed interval A :¼ ½0; 1Þ is continuous but not bounded on A. (ii) The interval must be closed. The function gðxÞ :¼ 1=x for x in the half-open interval B :¼ ð0; 1 is continuous but not bounded on B. (iii) The function must be continuous. The function h defined on the closed interval C :¼ ½0; 1 by hðxÞ :¼ 1=x for x 2 ð0; 1 and hð0Þ :¼ 1 is discontinuous and unbounded on C. The Maximum-Minimum Theorem 5.3.3 Definition Let A R and let f : A ! R. We say that f has an absolute maximum on A if there is a point x 2 A such that f ðx Þ f ðxÞ for all x 2 A: We say that f has an absolute minimum on A if there is a point x 2 A such that f ðx Þ f ðxÞ for all x 2 A: We say that x is an absolute maximum point for f on A, and that x is an absolute minimum point for f on A, if they exist. We note that a continuous function on a set A does not necessarily have an absolute maximum or an absolute minimum on the set. For example, f ðxÞ :¼ 1=x has neither an absolute maximum nor an absolute minimum on the set A :¼ ð0; 1Þ. (See Figure 5.3.1.) There can be no absolute maximum for f on A since f is not bounded above on A, and there is no point at which f attains the value 0 ¼ inff f ðxÞ : x 2 Ag. The same function has y This theorem, as well as 5.3.4, is true for an arbitrary closed bounded set. For these developments, see Sections 11.2 and 11.3. C05 12/08/2010 14:19:45 136 Page 136 CHAPTER 5 CONTINUOUS FUNCTIONS Figure 5.3.1 The function f ðxÞ ¼ 1=x ðx > 0Þ Figure 5.3.2 gðxÞ ¼ x2 The function ðjxj 1Þ neither an absolute maximum nor an absolute minimum when it is restricted to the set (0, 1), while it has both an absolute maximum and an absolute minimum when it is restricted to the set [1, 2]. In addition, f ðxÞ ¼ 1=x has an absolute maximum but no absolute minimum when restricted to the set [1, 1), but no absolute maximum and no absolute minimum when restricted to the set (1, 1). It is readily seen that if a function has an absolute maximum point, then this point is not necessarily uniquely determined. For example, the function gðxÞ :¼ x2 defined for x 2 A :¼ ½1; þ1 has the two points x ¼ 1 giving the absolute maximum on A, and the single point x ¼ 0 yielding its absolute minimum on A. (See Figure 5.3.2.) To pick an extreme example, the constant function hðxÞ :¼ 1 for x 2 R is such that every point of R is both an absolute maximum and an absolute minimum point for h. 5.3.4 Maximum-Minimum Theorem Let I :¼ ½a; b be a closed bounded interval and let f : I ! R be continuous on I. Then f has an absolute maximum and an absolute minimum on I. Proof. Consider the nonempty set f ðIÞ :¼ ff ðxÞ : x 2 Ig of values of f on I. In Theorem 5.3.2 it was established that f (I ) is a bounded subset of R. Let s :¼ sup f ðIÞ and s :¼ inf f ðI Þ. We claim that there exist points x and x in I such that s ¼ f ðx Þ and s ¼ f ðx Þ. We will establish the existence of the point x , leaving the proof of the existence of x to the reader. Since s ¼ sup f ðIÞ, if n 2 N, then the number s 1=n is not an upper bound of the set f (I). Consequently there exists a number xn 2 I such that ð1Þ 1 s < f ðxn Þ s n for all n 2 N: Since I is bounded, the sequence X :¼ ðxn Þ is bounded. Therefore, by the BolzanoWeierstrass Theorem 3.4.8, there is a subsequence X 0 ¼ ðxnr Þ of X that converges to some number x . Since the elements of X0 belong to I ¼ ½a; b, it follows from Theorem 3.2.6 that x 2 I. Therefore f is continuous at x so that limð f ðxnr ÞÞ ¼ f ðx Þ. Since it follows from (1) that s 1 < f ðxnr Þ s nr for all r 2 N; we conclude from the Squeeze Theorem 3.2.7 that limð f ðxnr ÞÞ ¼ s . Therefore we have f ðx Þ ¼ limð f ðxnr ÞÞ ¼ s ¼ sup f ðIÞ: We conclude that x is an absolute maximum point of f on I. Q.E.D. C05 12/08/2010 14:19:48 Page 137 5.3 CONTINUOUS FUNCTIONS ON INTERVALS 137 The next result is the theoretical basis for locating roots of a continuous function by means of sign changes of the function. The proof also provides an algorithm, known as the Bisection Method, for the calculation of roots to a specified degree of accuracy and can be readily programmed for a computer. It is a standard tool for finding solutions of equations of the form f (x) ¼ 0, where f is a continuous function. An alternative proof of the theorem is indicated in Exercise 5.3.11. 5.3.5 Location of Roots Theorem Let I ¼ ½a; b and let f : I ! R be continuous on I. If f ðaÞ < 0 < f ðbÞ, or if f ðaÞ > 0 > f ðbÞ, then there exists a number c 2 ða; bÞ such that f ðcÞ ¼ 0. Proof. We assume that f ðaÞ < 0 < f ðbÞ. We will generate a sequence of intervals by successive bisections. Let I 1 :¼ ½a1 ; b1 , where a1 :¼ a; b1 :¼ b, and let p1 be the midpoint p1 :¼ 12 ða1 þ b1 Þ. If f ðp1 Þ ¼ 0, we take c :¼ p1 and we are done. If f ðp1 Þ 6¼ 0, then either f ðp1 Þ > 0 or f ðp1 Þ < 0. If f ðp1 Þ > 0, then we set a2 :¼ a1 ; b2 :¼ p1 , while if f ðp1 Þ < 0, then we set a2 :¼ p1 ; b2 :¼ b1 . In either case, we let I 2 :¼ ½a2 ; b2 ; then we have I 2 I 1 and f ða2 Þ < 0; f ðb2 Þ > 0. We continue the bisection process. Suppose that the intervals I 1 ; I 2 ; . . . ; I k have been obtained by successive bisection in the same manner. Then we have f ðak Þ < 0 and f ðbk Þ > 0, and we set pk :¼ 12 ðak þ bk Þ. If f ðpk Þ ¼ 0, we take c :¼ pk and we are done. If f ðpk Þ > 0, we set akþ1 :¼ ak ; bkþ1 :¼ pk , while if f ðpk Þ < 0, we set akþ1 :¼ pk ; bkþ1 :¼ bk . In either case, we let I kþ1 :¼ ½akþ1 ; bkþ1 ; then I kþ1 I k and f ðakþ1 Þ < 0; f ðbkþ1 Þ > 0. If the process terminates by locating a point pn such that f ðpn Þ ¼ 0, then we are done. If the process does not terminate, then we obtain a nested sequence of closed bounded intervals I n :¼ ½an ; bn such that for every n 2 N we have f ðan Þ < 0 and f ðbn Þ > 0: Furthermore, since the intervals are obtained by repeated bisection, the length of In is equal to bn an ¼ ðb aÞ=2n1 . It follows from the Nested Intervals Property 2.5.2 that there exists a point c that belongs to In for all n 2 N. Since an c bn for all n 2 N and limðbn an Þ ¼ 0, it follows that limðan Þ ¼ c ¼ limðbn Þ. Since f is continuous at c, we have lim ð f ðan ÞÞ ¼ f ðcÞ ¼ lim ð f ðbn ÞÞ: The fact that f ðan Þ < 0 for all n 2 N implies that f ðcÞ ¼ lim ð f ðan ÞÞ 0. Also, the fact that f ðbn Þ > 0 for all n 2 N implies that f ðcÞ ¼ lim ð f ðbn ÞÞ 0. Thus, we conclude that f ðcÞ ¼ 0. Consequently, c is a root of f. Q.E.D. The following example illustrates how the Bisection Method for finding roots is applied in a systematic fashion. 5.3.6 Example The equation f ðxÞ ¼ xex 2 ¼ 0 has a root c in the interval [0, 1], because f is continuous on this interval and f ð0Þ ¼ 2 < 0 and f ð1Þ ¼ e 2 > 0. Using a calculator we construct the following table, where the sign of f ( pn) determines the interval at the next step. The far right column is an upper bound on the error when pn is used to approximate the root c, because we have jpn cj 12 ðbn an Þ ¼ 1=2n : C05 12/08/2010 14:19:49 138 Page 138 CHAPTER 5 CONTINUOUS FUNCTIONS We will find an approximation pn with error less than 102. n 1 2 3 4 5 6 7 an 0 .5 .75 .75 .8125 .84375 .84375 bn 1 1 1 .875 .875 .875 .859375 pn .5 .75 .875 .8125 .84375 .859375 .8515625 f ( pn) 1.176 .412 þ.099 .169 .0382 þ.0296 — 1 2 ðbn an Þ .5 .25 .125 .0625 .03125 .015625 .0078125 We have stopped at n ¼ 7, obtaining c p7 ¼ :8515625 with error less than .0078125. This is the first step in which the error is less than 102. The decimal place values of p7 past the second place cannot be taken seriously, but we can conclude that & :843 < c < :860. Bolzano’s Theorem The next result is a generalization of the Location of Roots Theorem. It assures us that a continuous function on an interval takes on (at least once) any number that lies between two of its values. 5.3.7 Bolzano’s Intermediate Value Theorem Let I be an interval and let f : I ! R be continuous on I. If a; b 2 I and if k 2 R satisfies f ðaÞ < k < f ðbÞ, then there exists a point c 2 I between a and b such that f ðcÞ ¼ k. Proof. Suppose that a < b and let gðxÞ :¼ f ðxÞ k; then gðaÞ < 0 < gðbÞ. By the Location of Roots Theorem 5.3.5 there exists a point c with a < c < b such that 0 ¼ gðcÞ ¼ f ðcÞ k. Therefore f ðcÞ ¼ k. If b < a, let hðxÞ :¼ k f ðxÞ so that hðbÞ < 0 < hðaÞ. Therefore there exists a point c with b < c < a such that 0 ¼ hðcÞ ¼ k f ðcÞ, whence f ðcÞ ¼ k. Q.E.D. 5.3.8 Corollary Let I ¼ ½a; b be a closed, bounded interval and let f : I ! R be continuous on I. If k 2 R is any number satisfying inf f ðIÞ k sup f ðIÞ; then there exists a number c 2 I such that f ðcÞ ¼ k. Proof. It follows from the Maximum-Minimum Theorem 5.3.4 that there are points c and c in I such that inf f ðIÞ ¼ f ðc Þ k f ðc Þ ¼ sup f ðIÞ: The conclusion now follows from Bolzano’s Theorem 5.3.7. Q.E.D. The next theorem summarizes the main results of this section. It states that the image of a closed bounded interval under a continuous function is also a closed bounded interval. The endpoints of the image interval are the absolute minimum and absolute maximum values of the function, and the statement that all values between the absolute minimum and the absolute maximum values belong to the image is a way of describing Bolzano’s Intermediate Value Theorem. C05 12/08/2010 14:19:50 Page 139 5.3 CONTINUOUS FUNCTIONS ON INTERVALS 139 5.3.9 Theorem Let I be a closed bounded interval and let f : I ! R be continuous on I. Then the set f ðIÞ :¼ f f ðxÞ : x 2 Ig is a closed bounded interval. Proof. If we let m :¼ inf f ðIÞ and M :¼ sup f ðIÞ, then we know from the MaximumMinimum Theorem 5.3.4 that m and M belong to f (I). Moreover, we have f ðIÞ ½m; M. If k is any element of [m, M], then it follows from the preceding corollary that there exists a point c 2 I such that k ¼ f ðcÞ. Hence, k 2 f ðIÞ and we conclude that ½m; M f ðIÞ. Therefore, f (I) is the interval [m, M]. Q.E.D. Warning If I :¼ ½a; b is an interval and f : I ! R is continuous on I, we have proved that f (I) is the interval [m, M]. We have not proved (and it is not always true) that f (I) is the & interval ½ f ðaÞ; f ðbÞ. (See Figure 5.3.3.) Figure 5.3.3 f ðIÞ ¼ ½m; M The preceding theorem is a ‘‘preservation’’ theorem in the sense that it states that the continuous image of a closed bounded interval is a set of the same type. The next theorem extends this result to general intervals. However, it should be noted that although the continuous image of an interval is shown to be an interval, it is not true that the image interval necessarily has the same form as the domain interval. For example, the continuous image of an open interval need not be an open interval, and the continuous image of an unbounded closed interval need not be a closed interval. Indeed, if f ðxÞ :¼ 1=ðx2 þ 1Þ for x 2 R, then f is continuous on R [see Example 5.2.3(b)]. It is easy to see that if I 1 :¼ ð1; 1Þ, then f ðI 1 Þ ¼ 12 ; 1 , which is not an open interval. Also, if I 2 :¼ ½0; 1Þ, then f ðI 2 Þ ¼ ð0; 1, which is not a closed interval. (See Figure 5.3.4.) Figure 5.3.4 Graph of f ðxÞ ¼ 1=ðx2 þ 1Þ ðx 2 RÞ C05 12/08/2010 14:19:53 140 Page 140 CHAPTER 5 CONTINUOUS FUNCTIONS To prove the Preservation of Intervals Theorem 5.3.10, we will use Theorem 2.5.1 characterizing intervals. 5.3.10 Preservation of Intervals Theorem Let I be an interval and let f : I ! R be continuous on I. Then the set f (I) is an interval. Proof. Let a; b 2 f ðIÞ with a < b; then there exist points a; b 2 I such that a ¼ f ðaÞ and b ¼ f ðbÞ. Further, it follows from Bolzano’s Intermediate Value Theorem 5.3.7 that if k 2 ða; bÞ then there exists a number c 2 I with k ¼ f ðcÞ 2 f ðIÞ. Therefore ½a; b f ðIÞ, showing that f (I ) possesses property (1) of Theorem 2.5.1. Therefore f (I ) is an interval. Q.E.D. Exercises for Section 5.3 1. Let I :¼ ½a; b and let f : I ! R be a continuous function such that f ðxÞ > 0 for each x in I. Prove that there exists a number a > 0 such that f ðxÞ a for all x 2 I. 2. Let I :¼ ½a; b and let f : I ! R and g : I ! R be continuous functions on I. Show that the set E :¼ fx 2 I : f ðxÞ ¼ gðxÞg has the property that if ðxn Þ E and xn ! x0 , then x0 2 E. 3. Let I :¼ ½a; b and let f : I ! R be a continuous function on I such that for each x in I there exists y in I such that j f ðyÞj 12 j f ðxÞj. Prove there exists a point c in I such that f ðcÞ ¼ 0. 4. Show that every polynomial of odd degree with real coefficients has at least one real root. 5. Show that the polynomial pðxÞ :¼ x4 þ 7x3 9 has at least two real roots. Use a calculator to locate these roots to within two decimal places. 6. Let f be continuous on the interval [0, 1] to R and such that f ð0Þ ¼ f ð1Þ. Prove that there exists a point c in [0, 12] such that f ðcÞ ¼ f c þ 12 . [Hint: Consider gðxÞ ¼ f ðxÞ f x þ 12 .] Conclude that there are, at any time, antipodal points on the earth’s equator that have the same temperature. 7. Show that the equation x ¼ cos x has a solution in the interval ½0; p=2. Use the Bisection Method and a calculator to find an approximate solution of this equation, with error less than 103. pffiffiffi 8. Show that the function f ðxÞ :¼ 2 ln x þ x 2 has root in the interval [1, 2], Use the Bisection Method and a calculator to find the root with error less than 102. 9. (a) The function f ðxÞ :¼ ðx 1Þðx 2Þðx 3Þðx 4Þðx 5Þ has five roots in the interval [0, 7]. If the Bisection Method is applied on this interval, which of the roots is located? (b) Same question for gðxÞ :¼ ðx 2Þðx 3Þðx 4Þðx 5Þðx 6Þ on the interval [0, 7]. 10. If the Bisection Method is used on an interval of length 1 to find pn with error j pn cj < 105, determine the least value of n that will assure this accuracy. 11. Let I :¼ ½a; b, let f : I ! R be continuous on I, and assume that f ðaÞ < 0; f ðbÞ > 0. Let W :¼ fx 2 I : f ðxÞ < 0g, and let w :¼ sup W. Prove that f ðwÞ ¼ 0. (This provides an alternative proof of Theorem 5.3.5.) 12. Let I :¼ ½0; p=2 and let f : I ! R be defined by f ðxÞ :¼ supfx2 ; cos xg for x 2 I. Show there exists an absolute minimum point x0 2 I for f on I. Show that x0 is a solution to the equation cos x ¼ x2 . 13. Suppose that f : R ! R is continuous on R and that lim f ¼ 0 and lim f ¼ 0. Prove that f is x!1 x!1 bounded on R and attains either a maximum or minimum on R. Give an example to show that both a maximum and a minimum need not be attained. 14. Let f : R ! R be continuous on R and let b 2 R. Show that if x0 2 R is such that f ðx0 Þ < b, then there exists a d-neighborhood U of x0 such that f ðxÞ < b for all x 2 U. C05 12/08/2010 14:19:54 Page 141 5.4 UNIFORM CONTINUITY 141 15. Examine which open [respectively, closed] intervals are mapped by f ðxÞ :¼ x2 for x 2 R onto open [respectively, closed] intervals. 16. Examine the mapping of open [respectively, closed] intervals under the functions gðxÞ :¼ 1=ðx2 þ 1Þ and hðxÞ :¼ x3 for x 2 R. 17. If f : ½0; 1 ! R is continuous and has only rational [respectively, irrational] values, must f be constant? Prove your assertion. 18. Let I :¼ ½a; b and let f : I ! R be a (not necessarily continuous) function with the property that for every x 2 I, the function f is bounded on a neighborhood V dx ðxÞ of x (in the sense of Definition 4.2.1). Prove that f is bounded on I. 19. Let J :¼ ða; bÞ and let g : J ! R be a continuous function with the property that for every x 2 J, the function g is bounded on a neighborhood V dx ðxÞ of x. Show by example that g is not necessarily bounded on J. Section 5.4 Uniform Continuity Let A R and let f : A ! R. Definition 5.1.1 states that the following statements are equivalent: (i) f is continuous at every point u 2 A; (ii) given e > 0 and u 2 A, there is a dðe; uÞ > 0 such that for all x such that x 2 A and jx uj < dðe; uÞ, then j f ðxÞ f ðuÞj < e. The point we wish to emphasize here is that d depends, in general, on both e > 0 and u 2 A. The fact that d depends on u is a reflection of the fact that the function f may change its values rapidly near certain points and slowly near other points. [For example, consider f ðxÞ :¼ sinð1=xÞ for x > 0; see Figure 4.1.3.] Now it often happens that the function f is such that the number d can be chosen to be independent of the point u 2 A and to depend only on e. For example, if f ðxÞ :¼ 2x for all x 2 R, then j f ðxÞ f ðuÞj ¼ 2jx uj; and so we can choose dðe; uÞ :¼ e=2 for all e > 0 and all u 2 R. (Why?) On the other hand if gðxÞ :¼ 1=x for x 2 A :¼ fx 2 R : x > 0g, then ux : ð1Þ gðxÞ gðuÞ ¼ ux If u 2 A is given and if we take ð2Þ dðe; uÞ :¼ inf 1 1 2 2 u; 2 u e ; then if jx uj < dðe; uÞ, we have jx uj < 12 u so that 12 u < x < 32 u, whence it follows that 1=x < 2=u. Thus, if jx uj < 12 u, the equality (1) yields the inequality ð3Þ jgðxÞ gðuÞj ð2=u2 Þjx uj: Consequently, if jx uj < dðe; uÞ, then (2) and (3) imply that jgðxÞ gðuÞj < ð2=u2 Þ 12 u2 e ¼ e: We have seen that the selection of dðe; uÞ by the formula (2) ‘‘works’’ in the sense that it enables us to give a value of d that will ensure that jgðxÞ gðuÞj < e when jx uj < d and C05 12/08/2010 14:19:55 142 Page 142 CHAPTER 5 CONTINUOUS FUNCTIONS x; u 2 A. We note that the value of dðe; uÞ given in (2) certainly depends on the point u 2 A. If we wish to consider all u 2 A, formula (2) does not lead to one value dðeÞ > 0 that will ‘‘work’’ simultaneously for all u > 0, since inffdðe; uÞ : u > 0g ¼ 0. In fact, there is no way of choosing one value of d that will ‘‘work’’ for all u > 0 for the function gðxÞ ¼ 1=x. The situation isexhibited graphically in Figures 5.4.1 and 5.4.2 where, for a given e-neighborhood V e 12 about 12 ¼ f ð2Þ and V e ð2Þ about 2 ¼ f 12 , the corresponding maximum values of d are seen to be considerably different. As u tends to 0, the permissible values of d tend to 0. Figure 5.4.1 gðxÞ ¼ 1=x ðx > 0Þ Figure 5.4.2 gðxÞ ¼ 1=x ðx > 0Þ 5.4.1 Definition Let A R and let f : A ! R. We say that f is uniformly continuous on A if for each e > 0 there is a dðeÞ > 0 such that if x; u 2 A are any numbers satisfying jx uj < dðeÞ, then j f ðxÞ f ðuÞj < e. It is clear that if f is uniformly continuous on A, then it is continuous at every point of A. In general, however, the converse does not hold, as is shown by the function gðxÞ ¼ 1=x on the set A :¼ fx 2 R : x > 0g. It is useful to formulate a condition equivalent to saying that f is not uniformly continuous on A. We give such criteria in the next result, leaving the proof to the reader as an exercise. 5.4.2 Nonuniform Continuity Criteria Let A R and let f : A ! R. Then the following statements are equivalent: (i) (ii) (iii) f is not uniformly continuous on A. There exists an e0 > 0 such that for every d > 0 there are points xd ; ud in A such that jxd ud j < d and j f ðxd Þ f ðud Þj e0 . There exists an e0 > 0 and two sequences (xn) and (un) in A such that limðxn un Þ ¼ 0 and j f ðxn Þ f ðun Þj e0 for all n 2 N. We can apply this result to show that gðxÞ :¼ 1=x is not uniformly continuous on A :¼ fx 2 R : x > 0g. For, if xn :¼ 1=n and un :¼ 1=ðn þ 1Þ, then we have limðxn un Þ ¼ 0, but jgðxn Þ gðun Þj ¼ 1 for all n 2 N. We now present an important result that assures that a continuous function on a closed bounded interval I is uniformly continuous on I. Other proofs of this theorem are given in Sections 5.5 and 11.3. C05 12/08/2010 14:19:57 Page 143 5.4 UNIFORM CONTINUITY 143 5.4.3 Uniform Continuity Theorem Let I be a closed bounded interval and let f : I ! R be continuous on I. Then f is uniformly continuous on I. Proof. If f is not uniformly continuous on I then, by the preceding result, there exists e0 > 0 and two sequences (xn) and (un) in I such that jxn un j < 1=n and j f ðxn Þ f ðun Þj e0 for all n 2 N. Since I is bounded, the sequence (xn) is bounded; by the BolzanoWeierstrass Theorem 3.4.8 there is a subsequence ðxnk Þ of (xn) that converges to an element z. Since I is closed, the limit z belongs to I, by Theorem 3.2.6. It is clear that the corresponding subsequence ðunk Þ also converges to z, since junk zj junk xnk j þ jxnk zj: Now if f is continuous at the point z, then both of the sequences ð f ðxnk ÞÞ and ð f ðunk ÞÞ must converge to f ðzÞ. But this is not possible since j f ðxn Þ f ðun Þj e0 for all n 2 N. Thus the hypothesis that f is not uniformly continuous on the closed bounded interval I implies that f is not continuous at some point z 2 I. Consequently, if f is continuous at every point of I, then f is uniformly continuous on I. Q.E.D. Lipschitz Functions If a uniformly continuous function is given on a set that is not a closed bounded interval, then it is sometimes difficult to establish its uniform continuity. However, there is a condition that frequently occurs that is sufficient to guarantee uniform continuity. It is named after Rudolf Lipschitz (1832–1903) who was a student of Dirichlet and who worked extensively in differential equations and Riemannian geometry. 5.4.4 Definition Let A R and let f : A ! R. If there exists a constant K > 0 such that ð4Þ j f ðxÞ f ðuÞj Kjx uj for all x; u 2 A, then f is said to be a Lipschitz function (or to satisfy a Lipschitz condition) on A. The condition (4) that a function f : I ! R on an interval I is a Lipschitz function can be interpreted geometrically as follows. If we write the condition as f ðxÞ f ðuÞ K; xu x; u 2 I; x 6¼ u ; then the quantity inside the absolute values is the slope of a line segment joining the points ðx; f ðxÞÞ and ðu; f ðuÞÞ. Thus a function f satisfies a Lipschitz condition if and only if the slopes of all line segments joining two points on the graph of y ¼ f ðxÞ over I are bounded by some number K. 5.4.5 Theorem If f : A ! R is a Lipschitz function, then f is uniformly continuous on A. Proof. If condition (4) is satisfied, then given e > 0, we can take d :¼ e=K. If x; u 2 A satisfy jx uj < d, then e j f ðxÞ f ðuÞj < K ¼ e: K Therefore f is uniformly continuous on A. Q.E.D. C05 12/08/2010 14:19:58 144 Page 144 CHAPTER 5 CONTINUOUS FUNCTIONS 5.4.6 Examples (a) If f ðxÞ :¼ x2 on A :¼ ½0; b, where b > 0, then j f ðxÞ f ðuÞj ¼ jx þ ujjx uj 2bjx uj for all x, u in [0, b]. Thus f satisfies (4) with K :¼ 2b on A, and therefore f is uniformly continuous on A. Of course, since f is continuous and A is a closed bounded interval, this can also be deduced from the Uniform Continuity Theorem. (Note that f does not satisfy a Lipschitz condition on the interval ½0; 1Þ.) (b) Not every uniformly continuous function is a Lipschitz function. pffiffiffi Let gðxÞ :¼ x for x in the closed bounded interval I :¼ ½0; 2. Since g is continuous on I, it follows from the Uniform Continuity Theorem 5.4.3 that g is uniformly continuous on I. However, there is no number K > 0 such that jgðxÞj Kjxj for all x 2 I. (Why not?) Therefore, g is not a Lipschitz function on I. (c) The Uniform Continuity Theorem and Theorem 5.4.5 can sometimes be combined to establish the uniform continuity pffiffiffi of a function on a set. We consider gðxÞ :¼ x on the set A :¼ ½0; 1Þ. The uniform continuity of g on the interval I :¼ ½0; 2 follows from the Uniform Continuity Theorem as noted in (b). If J :¼ ½1; 1Þ, then if both x, u are in J, we have jgðxÞ gðuÞj ¼ pffiffiffi pffiffiffi jx uj x u ¼ pffiffiffi pffiffiffi 12 jx uj: xþ u Thus g is a Lipschitz function on J with constant K ¼ 12, and hence by Theorem 5.4.5, g is uniformly continuous on [1, 1). Since A ¼ I [ J, it follows [by taking dðeÞ :¼ inf f1; dI ðeÞ; dJ ðeÞg] that g is uniformly continuous on A. We leave the details to the & reader. The Continuous Extension Theorem We have seen examples of functions that are continuous but not uniformly continuous on open intervals; for example, the function f ðxÞ ¼ 1=x on the interval (0, 1). On the other hand, by the Uniform Continuity Theorem, a function that is continuous on a closed bounded interval is always uniformly continuous. So the question arises: Under what conditions is a function uniformly continuous on a bounded open interval? The answer reveals the strength of uniform continuity, for it will be shown that a function on (a, b) is uniformly continuous if and only if it can be defined at the endpoints to produce a function that is continuous on the closed interval. We first establish a result that is of interest in itself. 5.4.7 Theorem If f : A ! R is uniformly continuous on a subset A of R and if ðxn Þ is a Cauchy sequence in A, then ð f ðxn ÞÞ is a Cauchy sequence in R. Proof. Let (xn) be a Cauchy sequence in A, and let e > 0 be given. First choose d > 0 such that if x, u in A satisfy jx uj < d, then j f ðxÞ f ðuÞj < e. Since (xn) is a Cauchy sequence, there exists HðdÞ such that jxn xm j < d for all n; m > HðdÞ. By the choice of d, this implies that for n; m > HðdÞ, we have j f ðxn Þ f ðxm Þj < e. Therefore the sequence ð f ðxn ÞÞ is a Cauchy sequence. Q.E.D. The preceding result gives us an alternative way of seeing that f ðxÞ :¼ 1=x is not uniformly continuous on (0, 1). We note that the sequence given by xn :¼ 1=n in (0, 1) is a Cauchy sequence, but the image sequence, where f ðxn Þ ¼ n, is not a Cauchy sequence. C05 12/08/2010 14:19:59 Page 145 5.4 UNIFORM CONTINUITY 145 5.4.8 Continuous Extension Theorem A function f is uniformly continuous on the interval (a, b) if and only if it can be defined at the endpoints a and b such that the extended function is continuous on [a, b]. Proof. ð(Þ This direction is trivial. ð)Þ Suppose f is uniformly continuous on (a, b). We shall show how to extend f to a; the argument for b is similar. This is done by showing that x!c lim f ðxÞ ¼ L exists, and this is accomplished by using the sequential criterion for limits. If (xn) is a sequence in (a, b) with limðxn Þ ¼ a, then it is a Cauchy sequence, and by the preceding theorem, the sequence ð f ðxn ÞÞ is also a Cauchy sequence, and so is convergent by Theorem 3.5.5. Thus the limit limð f ðxn ÞÞ ¼ L exists. If (un) is any other sequence in (a, b) that converges to a, then limðun xn Þ ¼ a a ¼ 0, so by the uniform continuity of f we have limð f ðun ÞÞ ¼ limð f ðun Þ f ðxn ÞÞ þ limð f ðxn ÞÞ ¼ 0 þ L ¼ L: Since we get the same value L for every sequence converging to a, we infer from the sequential criterion for limits that f has limit L at a. If we define f ðaÞ :¼ L, then f is continuous at a. The same argument applies to b, so we conclude that f has a continuous extension to the interval [a, b]. Q.E.D. Since the limit of f ðxÞ :¼ sinð1=xÞ at 0 does not exist, we infer from the Continuous Extension Theorem that the function is not uniformly continuous on ð0; b for any b > 0. On the other hand, since lim x sinð1=xÞ ¼ 0 exists, the function gðxÞ :¼ x sinð1=xÞ is x!0 uniformly continuous on ð0; b for all b > 0. Approximationy In many applications it is important to be able to approximate continuous functions by functions of an elementary nature. Although there are a variety of definitions that can be used to make the word ‘‘approximate’’ more precise, one of the most natural (as well as one of the most important) is to require that, at every point of the given domain, the approximating function shall not differ from the given function by more than the preassigned error. 5.4.9 Definition A function s : ½a; b ! R is called a step function if [a, b] is the union of a finite number of nonoverlapping intervals I 1 ; I 2 ; . . . ; I n such that s is constant on each interval, that is, sðxÞ ¼ ck for all x 2 I k ; k ¼ 1; 2; . . . ; n. Thus a step function has only a finite number of distinct values. For example, the function s : ½2; 4 ! R defined by 8 0; 2 x < 1; > > > > 1; 1 x 0; > > < 1 0 < x < 12 ; 2; sðxÞ :¼ 1 3; > > 2 x < 1; > > 2; 1 x 3; > > : 2; 3 < x 4; is a step function. (See Figure 5.4.3.) y The rest of this section can be omitted on a first reading of this chapter. C05 12/08/2010 14:20:0 146 Page 146 CHAPTER 5 CONTINUOUS FUNCTIONS Figure 5.4.3 Graph of y ¼ s(x) We will now show that a continuous function on a closed bounded interval I can be approximated arbitrarily closely by step functions. 5.4.10 Theorem Let I be a closed bounded interval and let f : I ! R be continuous on I. If e > 0, then there exists a step function se : I ! R such that j f ðxÞ se ðxÞj < e for all x 2 I. Proof. Since (by the Uniform Continuity Theorem 5.4.3) the function f is uniformly continuous, it follows that given e > 0 there is a number dðeÞ > 0 such that if x; y 2 I and jx yj < dðeÞ, then j f ðxÞ f ð yÞj < e. Let I :¼ ½a; b and let m 2 N be sufficiently large so that h :¼ ðb aÞ=m < dðeÞ. We now divide I ¼ [a, b] into m disjoint intervals of length h; namely, I 1 :¼ ½a; a þ h, and I k :¼ ða þ ðk 1Þh; a þ khÞ for k ¼ 2, . . . , m. Since the length of each subinterval Ik is h < dðeÞ, the difference between any two values of f in Ik is less than e. We now define ð5Þ se ðxÞ :¼ f ða þ khÞ for x 2 Ik; k ¼ 1; . . . ; m; so that se is constant on each interval Ik. (In fact the value of se on Ik is the value of f at the right endpoint of Ik. See Figure 5.4.4.) Consequently if x 2 I k , then j f ðxÞ se ðxÞj ¼ j f ðxÞ f ða þ khÞj < e: Therefore we have j f ðxÞ se ðxÞj < e for all x 2 I. Figure 5.4.4 Approximation by step functions Q.E.D. C05 12/08/2010 14:20:2 Page 147 5.4 UNIFORM CONTINUITY 147 Note that the proof of the preceding theorem establishes somewhat more than was announced in the statement of the theorem. In fact, we have proved the following, more precise, assertion. 5.4.11 Corollary Let I :¼ ½a; b be a closed bounded interval and let f : I ! R be continuous on I. If e > 0, there exists a natural number m such that if we divide I into m disjoint intervals Ik having length h :¼ ðb aÞ=m, then the step function se defined in equation (5) satisfies j f ðxÞ se ðxÞj < e for all x 2 I. Step functions are extremely elementary in character, but they are not continuous (except in trivial cases). Since it is often desirable to approximate continuous functions by elementary continuous functions, we now shall show that we can approximate continuous functions by continuous piecewise linear functions. 5.4.12 Definition Let I :¼ ½a; b be an interval. Then a function g : I ! R is said to be piecewise linear on I if I is the union of a finite number of disjoint intervals I 1 ; . . . ; I m , such that the restriction of g to each interval Ik is a linear function. Remark It is evident that in order for a piecewise linear function g to be continuous on I, the line segments that form the graph of g must meet at the endpoints of adjacent subintervals I k ; I kþ1 ðk ¼ 1; . . . ; m 1Þ. 5.4.13 Theorem Let I be a closed bounded interval and let f : I ! R be continuous on I. If e > 0, then there exists a continuous piecewise linear function ge : I ! R such that j f ðxÞ ge ðxÞj < e for all x 2 I. Proof. Since f is uniformly continuous on I :¼ ½a; b, there is a number dðeÞ > 0 such that if x, y 2 I and jx yj < dðeÞ, then j f ðxÞ f ðyÞj < e. Let m 2 N be sufficiently large so that h :¼ ðb aÞ=m < dðeÞ. Divide I ¼ ½a; b into m disjoint intervals of length h; namely, let I 1 ¼ ½a; a þ h, and let I k ¼ ða þ ðk 1Þh; a þ kh for k ¼ 2; . . . ; m. On each interval Ik we define ge to be the linear function joining the points ða þ ðk 1Þh; f ða þ ðk 1ÞhÞÞ and ða þ kh; f ða þ khÞÞ : Then ge is a continuous piecewise linear function on I. Since, for x 2 I k the value f(x) is within e of f ða þ ðk 1ÞhÞ and f ða þ khÞ, it is an exercise to show that j f ðxÞ ge ðxÞj < e for all x 2 I k ; therefore this inequality holds for all x 2 I. (See Figure 5.4.5.) Q.E.D. Figure 5.4.5 Approximation by piecewise linear function C05 12/08/2010 14:20:3 148 Page 148 CHAPTER 5 CONTINUOUS FUNCTIONS We shall close this section by stating the important theorem of Weierstrass concerning the approximation of continuous functions by polynomial functions. As would be expected, in order to obtain an approximation within an arbitrarily preassigned e > 0, we must be prepared to use polynomials of arbitrarily high degree. 5.4.14 Weierstrass Approximation Theorem Let I ¼ ½a; b and let f : I ! R be a continuous function. If e > 0 is given, then there exists a polynomial function pe such that j f ðxÞ pe ðxÞj < e for all x 2 I. There are a number of proofs of this result. Unfortunately, all of them are rather intricate, or employ results that are not yet at our disposal. (A proof can be found in Bartle, ERA, pp. 169–172, which is listed in the References.) Exercises for Section 5.4 1. Show that the function f ðxÞ :¼ 1=x is uniformly continuous on the set A :¼ ½a; 1Þ, where a is a positive constant. 2. Show that the function f ðxÞ :¼ 1=x2 is uniformly continuous on A :¼ ½1; 1Þ, but that it is not uniformly continuous on B :¼ ð0; 1Þ. 3. Use the Nonuniform Continuity Criterion 5.4.2 to show that the following functions are not uniformly continuous on the given sets. (a) f ðxÞ :¼ x2 ; A :¼ ½0; 1Þ . (b) gðxÞ :¼ sinð1=xÞ; B :¼ ð0; 1Þ . 4. Show that the function f ðxÞ :¼ 1=ð1 þ x2 Þ for x 2 R is uniformly continuous on R. 5. Show that if f and g are uniformly continuous on a subset A of R, then f þ g is uniformly continuous on A. 6. Show that if f and g are uniformly continuous on A R and if they are both bounded on A, then their product fg is uniformly continuous on A. 7. If f ðxÞ :¼ x and gðxÞ :¼ sin x, show that both f and g are uniformly continuous on R, but that their product fg is not uniformly continuous on R. 8. Prove that if f and g are each uniformly continuous on R, then the composite function f g is uniformly continuous on R. 9. If f is uniformly continuous on A R, and j f ðxÞj k > 0 for all x 2 A, show that 1=f is uniformly continuous on A. 10. Prove that if f is uniformly continuous on a bounded subset A of R, then f is bounded on A. pffiffiffi 11. If gðxÞ :¼ x for x 2 ½0; 1, show that there does not exist a constant K such that jgðxÞj K jxj for all x 2 ½0; 1. Conclude that the uniformly continuous g is not a Lipschitz function on [0, 1]. 12. Show that if f is continuous on [0, 1) and uniformly continuous on [a, 1) for some positive constant a, then f is uniformly continuous on [0, 1). 13. Let A R and suppose that f : A ! R has the following property: for each e > 0 there exists a function ge : A ! R such that ge is uniformly continuous on A and j f ðxÞ ge ðxÞj < e for all x 2 A. Prove that f is uniformly continuous on A. 14. A function f : R ! R is said to be periodic on R if there exists a number p > 0 such that f ðx þ pÞ ¼ f ðxÞ for all x 2 R. Prove that a continuous periodic function on R is bounded and uniformly continuous on R. C05 12/08/2010 14:20:5 Page 149 5.5 CONTINUITY AND GAUGES 15. Let (a) (b) (c) 149 f and g be Lipschitz functions on A. Show that the sum f þ g is also a Lipschitz function on A. Show that if f and g are bounded on A, then the product fg is a Lipschitz function on A. Give an example of a Lipschitz function f on [0, 1) such that its square f 2 is not a Lipschitz function. 16. A function is called absolutely continuous on an interval I if for any e > 0 there exists a d > 0 such P that for any pair-wise P disjoint subintervals ½xk ; yk ; k ¼ 1; 2; . . . ; n, of I such that j f ðxk Þ f ðyk Þj < e. Show that if f satisfies a Lipschitz condition jxk yk j < d we have on I, then f is absolutely continuous on I. Section 5.5 Continuity and Gaugesy We will now introduce some concepts that will be used later—especially in Chapters 7 and 10 on integration theory. However, we wish to introduce the notion of a ‘‘gauge’’ now because of its connection with the study of continuous functions. We first define the notion of a tagged partition of an interval. 5.5.1 Definition A partition of an interval I :¼ ½a; b is a collection P ¼ fI 1 ; . . . ; I n g of non-overlapping closed intervals whose union is [a, b]. We ordinarily denote the intervals by I i :¼ ½xi1 ; xi , where a ¼ x0 < < xi1 < xi < < xn ¼ b: The points xi ði ¼ 0; . . . ; nÞ are called the partition points of P. If a point ti has been chosen from each interval Ii, for i ¼ 1, . . . , n, then the points ti are called the tags and the set of ordered pairs P ¼ fðI 1 ; t1 Þ; . . . ; ðI n ; tn Þg is called a tagged partition of I. (The dot signifies that the partition is tagged.) The ‘‘fineness’’ of a partition P refers to the lengths of the subintervals in P. Instead of requiring that all subintervals have length less than some specific quantity, it is often useful to allow varying degrees of fineness for different subintervals Ii in P. This is accomplished by the use of a ‘‘gauge,’’ which we now define. 5.5.2 Definition A gauge on I is a strictly positive function defined on I. If d is a gauge on I, then a (tagged) partition P_ is said to be d-fine if ð1Þ ti 2 I i ½ti dðti Þ; ti þ dðti Þ for i ¼ 1; . . . ; n : We note that the notion of d-fineness requires that the partition be tagged, so we do not need to say ‘‘tagged partition’’ in this case. A gauge d on an interval I assigns an interval ½t dðtÞ; t þ dðtÞ to each point t 2 I. The d-fineness of a partition P_ requires that each subinterval Ii of P_ is contained in the interval determined by the gauge d and the tag ti for that subinterval. This is indicated by the inclusions in (1); see Figure 5.5.1. Note that the length of the subintervals is also controlled by the gauge and the tags; the next lemma reflects that control. y This section can be omitted on a first reading of this chapter. C05 12/08/2010 14:20:5 150 Page 150 CHAPTER 5 CONTINUOUS FUNCTIONS Figure 5.5.1 Inclusion (1) 5.5.3 Lemma If a partition P_ of I :¼ ½a; b is d-fine and x 2 I, then there exists a tag ti in P_ such that jx ti j dðti Þ. Proof. If x 2 I, there exists a subinterval ½xi1 ; xi from P_ that contains x. Since P_ is d-fine, then ti dðti Þ xi1 x xi ti þ dðti Þ; ð2Þ whence it follows that jx ti j dðti Þ. Q.E.D. In the theory of Riemann integration, we will use gauges d that are constant functions to control the fineness of the partition; in the theory of the generalized Riemann integral, the use of nonconstant gauges is essential. But nonconstant gauge functions arise quite naturally in connection with continuous functions. For, let f : I ! R be continuous on I and let e > 0 be given. Then, for each point t 2 I there exists de ðtÞ > 0 such that if jx tj < de ðtÞ and x 2 I, then j f ðxÞ f ðtÞj < e. Since de is defined and is strictly positive on I, the function de is a gauge on I. Later in this section, we will use the relations between gauges and continuity to give alternative proofs of the fundamental properties of continuous functions discussed in Sections 5.3 and 5.4. 5.5.4 Examples (a) If d and g are gauges on I :¼ ½a; b and if 0 < dðxÞ g ðxÞ for all x 2 I, then every partition P_ that is d-fine is also g-fine. This follows immediately from the inequalities ti gðti Þ ti dðti Þ and ti þ dðti Þ ti þ gðti Þ which imply that ti 2 ½ti dðti Þ; ti þ dðti Þ ½ti gðti Þ; ti þ gðti Þ for i ¼ 1; . . . ; n : (b) If d1 and d2 are gauges on I :¼ ½a; b and if dðxÞ :¼ minfd1 ðxÞ; d2 ðxÞg for all x2I; then d is also a gauge on I. Moreover, since dðxÞ d1 ðxÞ, then every d-fine partition is d1 fine. Similarly, every d-fine partition is also d2 -fine. (c) Suppose that d is defined on I :¼ ½0; 1 by ( dðxÞ :¼ 1 10 if x ¼ 0; 1 2x if 0 < x 1: Then d is a gauge on [0, 1]. If 0 < t 1, then ½t dðtÞ; t þ dðtÞ ¼ 12 t; 32 t , which does not contain the point 0. Thus, if P_ is a d-fine partition of I, then the only subinterval in P_ that contains 0 must have the point 0 as its tag. C05 12/08/2010 14:20:7 Page 151 5.5 CONTINUITY AND GAUGES 151 (d) Let g be defined on I :¼ ½0; 1 by 81 > > < 10 gðxÞ :¼ 12 x > > :1 2 ð1 xÞ if x ¼ 0 or x ¼ 1; if 0 < x 12 ; if 1 2 < x < 1: Then g is a gauge on I, and it is an exercise to show that the subintervals in any g-fine partition that contain the points 0 or 1 must have these points as tags. & Existence of d-Fine Partitions In view of the above examples, it is not obvious that an arbitrary gauge d admits a d-fine partition. We now use the Supremum Property of R to establish the existence of d-fine partitions. In the exercises, we will sketch a proof based on the Nested Intervals Theorem 2.5.2. 5.5.5 Theorem If d is a gauge defined on the interval [a, b], then there exists a d-fine partition of [a, b]. Proof. Let E denote the set of all points x 2 ½a; b such that there exists a d-fine partition of the subinterval [a, x]. The set E is not empty, since the pair ([a, x], a) is a d-fine partition of the interval [a, x] when x 2 ½a; a þ dðaÞ and x b. Since E ½a; b, the set E is also bounded. Let u :¼ sup E so that a < u b. We will show that u 2 E and that u ¼ b. We claim that u 2 E. Since u dðuÞ < u ¼ sup E, there exists v 2 E such that u dðuÞ < v < u. Let P_ 1 be a d-fine partition of [a, v] and let P_ 2 :¼ P_ 1 [ ð½v; u; uÞ. Then P_ 2 is a d-fine partition of [a, u], so that u 2 E. _ 1 is a d-fine partition of If u < b, let w 2 ½a; b be such that u < w < u þ dðuÞ. If Q _ _ _ [a, u], we let Q2 :¼ Q1 [ ð½u; w; uÞ. Then Q2 is a d-fine partition of [a, w], whence w 2 E. But this contradicts the supposition that u is an upper bound of E. Therefore u ¼ b. Q.E.D. Some Applications Following R. A. Gordon (see his Monthly article in the References), we will now show that some of the major theorems in the two preceding sections can be proved by using gauges. Alternate Proof of Theorem 5.3.2: Boundedness Theorem. Since f is continuous on I, then for each t 2 I there exists dðtÞ > 0 such that if x 2 I and jx tj dðtÞ, then j f ðxÞ f ðtÞj 1. Thus d is a gauge on I. Let fðI i ; ti Þgni¼1 be a d-fine partition of I and let K :¼ maxfj f ðti Þj : i ¼ 1; . . . ; ng. By Lemma 5.5.3, given any x 2 I there exists i with jx ti j dðti Þ, whence j f ðxÞj j f ðxÞ f ðti Þj þ j f ðti Þj 1 þ K: Since x 2 I is arbitrary, then f is bounded by 1 þ K on I. Q.E.D. C05 12/08/2010 14:20:9 152 Page 152 CHAPTER 5 CONTINUOUS FUNCTIONS Alternate Proof of Theorem 5.3.4: Maximum-Minimum Theorem. We will prove the existence of x . Let M :¼ supff ðxÞ : x 2 I g and suppose that f(x) < M for all x 2 I. Since f is continuous on I, for each t 2 I there exists dðtÞ > 0 such that if x 2 I and jx tj dðtÞ, then f ðxÞ < 12 ðM þ f ðtÞÞ. Thus d is a gauge on I, and if fðI i ; ti Þgni¼1 is a d-fine partition of I, we let ~ M :¼ 12 maxfM þ f ðt1 Þ; . . . ; M þ f ðtn Þg: By Lemma 5.5.3, given any x 2 I, there exists i with jx ti j dðti Þ, whence ~ f ðxÞ < 12 ðM þ f ðti ÞÞ M: ~ Since x 2 I is arbitrary, then Mð< MÞ is an upper bound for f on I, contrary to the definition of M as the supremum of f. Q.E.D. Alternate Proof of Theorem 5.3.5: Location of Roots Theorem. We assume that f ðtÞ 6¼ 0 for all t 2 I. Since f is continuous at t, Exercise 5.1.7 implies that there exists dðtÞ > 0 such that if x 2 I and jx tj dðtÞ, then f (x) < 0 if f (t) < 0, and f (x) > 0 if f (t) > 0. Then d is a gauge on I and we let fðI i ; ti Þgni¼1 be a d-fine partition. Note that for each i, either f (x) < 0 for all x 2 ½xi1 ; xi or f (x) > 0 for all such x. Since f ðx0 Þ ¼ f ðaÞ < 0, this implies that f (x1) < 0, which in turn implies that f (x2) < 0. Continuing in this way, we have f ðbÞ ¼ f ðxn Þ < 0, contrary to the hypothesis that f (b) > 0. Q.E.D. Alternate Proof of Theorem 5.4.3: Uniform Continuity Theorem. Let e > 0 be given. Since f is continuous at t 2 I, there exists dðtÞ > 0 such that if x 2 I and jx tj 2dðtÞ, then j f ðxÞ f ðtÞj 12 e. Thus d is a gauge on I. If fðI i ; ti Þgni¼1 is a d-fine partition of I, let de :¼ minfdðt1 Þ; . . . ; dðtn Þg. Now suppose that x; u 2 I and jx uj de , and choose i with jx ti j dðti Þ. Since ju ti j ju xj þ jx ti j de þ dðti Þ 2dðti Þ; then it follows that j f ðxÞ f ðuÞj j f ðxÞ f ðti Þj þ j f ðti Þ f ðuÞj 12 e þ 12 e ¼ e: Therefore, f is uniformly continuous on I. Q.E.D. Exercises for Section 5.5 1. Let d be the gauge on [0, 1] defined by dð0Þ: ¼ 14 and dðtÞ :¼ 12 t for t 2 ð0; 1. 1 (a) Show that P_ 1 ¼ 0; 4 ; 0 ; 14 ; 12 ; 12 ; 12 ; 1 ; 34 is d-fine. 1 1 1 1 1 3 (b) Show that P_ 2 ¼ 0; 4 ; 0 ; 4 ; 2 ; 2 ; 2 ; 1 ; 5 is not d-fine. 2. Suppose that d1 is the gauge defined by d1 ð0Þ :¼ 14 ; d1 ðtÞ :¼ 34 t for t 2 ð0; 1. Are the partitions given in Exercise 1 d1 -fine? Note that dðtÞ d1 ðtÞ for all t 2 ½0; 1. 1 9 3. Suppose that d2 is the gauge defined by d2 ð0Þ :¼ 10 and d2 ðtÞ :¼ 10 t for t 2 ð0; 1. Are the partitions given in Exercise 1 d2 -fine? 4. Let g be the gauge in Example 5.5.4(d). (a) If t 2 ð0; 12 show that ½t gðtÞ; t þ gðtÞ ¼ 12 t; 32 t ð0; 34. (b) If t 2 ð12 ; 1Þ show that ½t gðtÞ; t þ gðtÞ ð14 ; 1Þ. 5. Let a < c < b and let d be a gauge on [a, b]. If P_ 0 is a d-fine partition of [a, c] and if P_00 is a d-fine partition of [c, b], show that P_ 0 [ P_00 is a d-fine partition of [a, b] having c as a partition point. C05 12/08/2010 14:20:10 Page 153 5.6 MONOTONE AND INVERSE FUNCTIONS 153 6. Let a < c < b and let d0 and d00 be gauges on [a, c] and [c, b], respectively. If d is defined on [a, b] by 80 d ðtÞ > > < dðtÞ :¼ minfd0 ðcÞ; d00 ðcÞg > > : 00 d ðtÞ if t 2 ½a; cÞ; if t ¼ c; if t 2 ðc; b; 00 then d is a gauge on [a, b]. Moreover, if P_ 0 is a d0 -fine partition of [a, c] and P_00 is a d -fine partition of [c, b], then P_ 0 [ P_00 is a tagged partition of [a, b] having c as a partition point. Explain why P_ 0 [ P_00 may not be d-fine. Give an example. 00 7. Let d0 and d be as in the preceding exercise and let d be defined by 8 min d0 ðtÞ; 12 ðc tÞ > > < d ðtÞ :¼ minfd0 ðcÞ; d00 ðcÞg > > : min d00 ðtÞ; 12 ðt cÞ if t 2 ½a; cÞ; if t ¼ c; if t 2 ðc; b: Show that d is a gauge on [a, b] and that every d -fine partition P_ of [a, b] having c as a partition point gives rise to a d0 -fine partition P_ 0 of [a, c] and a d00 -fine partition P_ 00 of [c, b] such that P_ ¼ P_ 0 [ P_00 . 8. Let d be a gauge on I :¼ ½a; b and suppose that I does not have a d-fine partition. (a) Let c :¼ 12 ða þ bÞ. Show that at least one of the intervals [a, c] and [c, b] does not have a d-fine partition. (b) Construct a nested sequence (In) of subintervals with the length of In equal to ðb aÞ=2n such that In does not have a d-fine partition. p (c) Let j 2 \1 n¼1 I n and let p 2 N be such that ðb aÞ=2 < dðjÞ. Show that I p ½j dðjÞ; j þ dðjÞ, so the pair I p ; j is a d-fine partition of Ip. 9. Let I :¼ ½a; b and let f : I ! R be a (not necessarily continuous) function. We say that f is ‘‘locally bounded’’ at c 2 I if there exists dðcÞ > 0 such that f is bounded on I \ ½c dðcÞ; c þ dðcÞ. Prove that if f is locally bounded at every point of I, then f is bounded on I. 10. Let I :¼ ½a; b and f : I ! R. We say that f is ‘‘locally increasing’’ at c 2 I if there exists dðcÞ > 0 such that f is increasing on I \ ½c dðcÞ; c þ dðcÞ. Prove that if f is locally increasing at every point of I, then f is increasing on I. Section 5.6 Monotone and Inverse Functions Recall that if A R, then a function f : A ! R is said to be increasing on A if whenever x1 ; x2 2 A and x1 x2 , then f ðx1 Þ f ðx2 Þ. The function f is said to be strictly increasing on A if whenever x1 ;x2 2 A and x1 < x2 , then f ðx1 Þ < f ðx2 Þ. Similarly, g : A ! R is said to be decreasing on A if whenever x1 ; x2 2 A and x1 x2 then gðx1 Þ gðx2 Þ. The function g is said to be strictly decreasing on A if whenever x1 ; x2 2 A and x1 < x2 then gðx1 Þ > gðx2 Þ. If a function is either increasing or decreasing on A, we say that it is monotone on A. If f is either strictly increasing or strictly decreasing on A, we say that f is strictly monotone on A. We note that if f : A ! R is increasing on A then g :¼ f is decreasing on A; similarly if w : A ! R is decreasing on A then c :¼ w is increasing on A. C05 12/08/2010 14:20:11 154 Page 154 CHAPTER 5 CONTINUOUS FUNCTIONS In this section, we will be concerned with monotone functions that are defined on an interval I R. We will discuss increasing functions explicitly, but it is clear that there are corresponding results for decreasing functions. These results can either be obtained directly from the results for increasing functions or proved by similar arguments. Monotone functions are not necessarily continuous. For example, if f ðxÞ :¼ 0 for x 2 ½0; 1 and f ðxÞ :¼ 1 for x 2 ð1; 2, then f is increasing on [0, 2], but fails to be continuous at x ¼ 1. However, the next result shows that a monotone function always has both one-sided limits (see Definition 4.3.1) in R at every point that is not an endpoint of its domain. 5.6.1 Theorem Let I R be an interval and let f : I ! R be increasing on I. Suppose that c 2 I is not an endpoint of I. Then (i) (ii) lim f ¼ supf f ðxÞ : x 2 I; x < cg, x!c lim f ¼ inf f f ðxÞ : x 2 I; x > cg. x!cþ Proof. (i) First note that if x 2 I and x < c, then f ðxÞ f ðcÞ. Hence the set f f ðxÞ : x 2 I; x < cg, which is nonvoid since c is not an endpoint of I, is bounded above by f (c). Thus the indicated supremum exists; we denote it by L. If e > 0 is given, then L e is not an upper bound of this set. Hence there exists ye 2 I; ye < c such that L e < f ðye Þ L. Since f is increasing, we deduce that if de :¼ c ye and if 0 < c y < de , then ye < y < c so that L e < f ðye Þ f ðyÞ L: Therefore j f ðyÞ Lj < e when 0 < c y < de . Since e > 0 is arbitrary we infer that (i) holds. The proof of (ii) is similar. Q.E.D. The next result gives criteria for the continuity of an increasing function f at a point c that is not an endpoint of the interval on which f is defined. 5.6.2 Corollary Let I R be an interval and let f : I ! R be increasing on I. Suppose that c 2 I is not an endpoint of I. Then the following statements are equivalent. (a) f is continuous at c. (b) lim f ¼ f ðcÞ ¼ lim f . x!c x!cþ (c) supf f ðxÞ : x 2 I; x < cg ¼ f ðcÞ ¼ inf f f ðxÞ : x 2 I; x > cg. This follows easily from Theorems 4.3.3 and 5.6.1. We leave the details to the reader. Let I be an interval and let f : I ! R be an increasing function. If a is the left endpoint of I, it is an exercise to show that f is continuous at a if and only if f ðaÞ ¼ inf f f ðxÞ : x 2 I; a < xg or if and only if f ðaÞ ¼ lim f . Similar conditions apply at a right endpoint, and for x!aþ decreasing functions. C05 12/08/2010 14:20:12 Page 155 5.6 MONOTONE AND INVERSE FUNCTIONS Figure 5.6.1 155 The jump of f at c If f : I ! R is increasing on I and if c is not an endpoint of I, we define the jump of f at c to be j f ðcÞ :¼ lim f lim f . (See Figure 5.6.1.) It follows from Theorem 5.6.1 x!cþ x!c that j f ðcÞ ¼ inf f f ðxÞ : x 2 I; x > cg supf f ðxÞ : x 2 I; x < cg for an increasing function. If the left endpoint a of I belongs to I, we define the jump of f at a to be j f ðaÞ :¼ lim f f ðaÞ. If the right endpoint b of I belongs to I, we define the jump x!aþ of f at b to be j f ðbÞ :¼ f ðbÞ lim f . x!b 5.6.3 Theorem Let I R be an interval and let f : I ! R be increasing on I. If c 2 I, then f is continuous at c if and only if j f ðcÞ ¼ 0. Proof. If c is not an endpoint, this follows immediately from Corollary 5.6.2. If c 2 I is the left endpoint of I, then f is continuous at c if and only if f ðcÞ ¼ lim f , which is x!cþ equivalent to j f ðcÞ ¼ 0. Similar remarks apply to the case of a right endpoint. Q.E.D. We now show that there can be at most a countable set of points at which a monotone function is discontinuous. 5.6.4 Theorem Let I R be an interval and let f : I ! R be monotone on I. Then the set of points D I at which f is discontinuous is a countable set. Proof. We shall suppose that f is increasing on I. It follows from Theorem 5.6.3 that D ¼ x 2 I : j f ð xÞ 6¼ 0 . We shall consider the case that I :¼ ½a; b is a closed bounded interval, leaving the case of an arbitrary interval to the reader. We first note that since f is increasing, then j f ðcÞ 0 for all c 2 I. Moreover, if a x1 < xn b, then (why?) we have ð1Þ f ðaÞ f ðaÞ þ j f ðx1 Þ þ þ j f ðxn Þ f ðbÞ; whence it follows that j f ðx1 Þ þ þ j f ðxn Þ f ðbÞ f ðaÞ: (See Figure 5.6.2.) Consequently there can be at most k points in I ¼ ½a; b where j f ðxÞ ð f ðbÞ f ðaÞÞ=k. We conclude that there is at most one point x 2 I where j f ðxÞ ¼ f ðbÞ f ðaÞ; there are at most two points in I where j f ðxÞ ð f ðbÞ f ðaÞÞ=2; at most three points in I where j f ðxÞ ð f ðbÞ f ðaÞÞ=3, and so on. Therefore there is at C05 12/08/2010 14:20:12 156 Page 156 CHAPTER 5 CONTINUOUS FUNCTIONS Figure 5.6.2 j f ðx1 Þ þ þ j f ðxn Þ f ðbÞ f ðaÞ most a countable set of points x where j f ðxÞ > 0. But since every point in D must be included in this set, we deduce that D is a countable set. Q.E.D. Theorem 5.6.4 has some useful applications. For example, it was seen in Exercise 5.2.12 that if h : R ! R satisfies the identity ð2Þ hðx þ yÞ ¼ hðxÞ þ hðyÞ for all x; y 2 R ; and if h is continuous at a single point x0, then h is continuous at every point of R. Thus, if h is a monotone function satisfying (2), then h must be continuous on R. [It follows from this that h(x) ¼ Cx for all x 2 R, where C :¼ hð1Þ.] Inverse Functions We shall now consider the existence of inverses for functions that are continuous on an interval I R. We recall (see Section 1.1) that a function f : I ! R has an inverse function if and only if f is injective (¼ one-one); that is, x; y 2 I and x 6¼ y imply that f ðxÞ 6¼ f ðyÞ. We note that a strictly monotone function is injective and so has an inverse. In the next theorem, we show that if f : I ! R is a strictly monotone continuous function, then f has an inverse function g on J :¼ f ðIÞ that is strictly monotone and continuous on J. In particular, if f is strictly increasing then so is g, and if f is strictly decreasing then so is g. 5.6.5 Continuous Inverse Theorem Let I R be an interval and let f : I ! R be strictly monotone and continuous on I. Then the function g inverse to f is strictly monotone and continuous on J :¼ f ðIÞ. Proof. We consider the case that f is strictly increasing, leaving the case that f is strictly decreasing to the reader. Since f is continuous and I is an interval, it follows from the Preservation of Intervals Theorem 5.3.10 that J :¼ f ðIÞ is an interval. Moreover, since f is strictly increasing on I, it is injective on I; therefore the function g : J ! R inverse to f exists. We claim that g is C05 12/08/2010 14:20:13 Page 157 5.6 MONOTONE AND INVERSE FUNCTIONS Figure 5.6.3 157 gðyÞ 6¼ x for y 2 J strictly increasing. Indeed, if y1 ; y2 2 J with y1 < y2 , then y1 ¼ f ðx1 Þ and y2 ¼ f ðx2 Þ for some x1 ; x2 2 I. We must have x1 < x2 ; otherwise x1 x2 , which implies that y1 ¼ f ðx1 Þ f ðx2 Þ ¼ y2 , contrary to the hypothesis that y1 < y2 . Therefore we have gðy1 Þ ¼ x1 < x2 ¼ gðy2 Þ. Since y1 and y2 are arbitrary elements of J with y1 < y2 , we conclude that g is strictly increasing on J. It remains to show that g is continuous on J. However, this is a consequence of the fact that gðJÞ ¼ I is an interval. Indeed, if g is discontinuous at a point c 2 J, then the jump of g at c is nonzero so that lim g < lim g. If we choose any number x 6¼ gðcÞ satisfying y!c y!cþ lim g < x < lim g, then x has the property that x 6¼ gðyÞ for any y 2 J. (See x!c x!cþ Figure 5.6.3.) Hence x 2 = I, which contradicts the fact that I is an interval. Therefore we conclude that g is continuous on J. Q.E.D. The nth Root Function We will apply the Continuous Inverse Theorem 5.6.5 to the nth power function. We need to distinguish two cases: (i) n even, and (ii) n odd. (i) n even. In order to obtain a function that is strictly monotone, we restrict our attention to the interval I :¼ ½0; 1Þ. Thus, let f ðxÞ :¼ xn for x 2 I. (See Figure 5.6.4.) We have seen (in Exercise 2.1.23) that if 0 x < y, then f ðxÞ ¼ xn < yn ¼ f ðyÞ; therefore f is strictly increasing on I. Moreover, it follows from Example 5.2.3(a) that f is continuous on I. Therefore, by the Preservation of Intervals Theorem 5.3.10, J :¼ f ðIÞ is an interval. We will show that J ¼ ½0; 1Þ. Let y 0 be arbitrary; by the Archimedean Property, there exists k 2 N such that 0 y < k. Since f ð0Þ ¼ 0 y < k kn ¼ f ðkÞ; it follows from Bolzano’s Intermediate Value Theorem 5.3.7 that y 2 J. Since y 0 is arbitrary, we deduce that J ¼ ½0; 1Þ. We conclude from the Continuous Inverse Theorem 5.6.5 that the function g that is inverse to f ðxÞ ¼ xn on I ¼ ½0; 1Þ is strictly increasing and continuous on J ¼ ½0; 1Þ. We usually write pffiffiffi gðxÞ ¼ x1=n or gðxÞ ¼ n x C05 12/08/2010 14:20:14 158 Page 158 CHAPTER 5 CONTINUOUS FUNCTIONS Figure 5.6.5 Graph of gðxÞ ¼ x1=n ðx 0; n evenÞ Figure 5.6.4 Graph of f ðxÞ ¼ xn ðx 0; n evenÞ pffiffiffi for x 0 (n even), and call x1=n ¼ n x the nth root of x 0 (n even). The function g is called the nth root function (n even). (See Figure 5.6.5.) Since g is inverse to f we have gð f ðxÞÞ ¼ x and f ðgðxÞÞ ¼ x for all x 2 ½0; 1Þ: We can write these equations in the following form: ðxn Þ1=n ¼ x and ðx1=n Þn ¼ x for all x 2 ½0; 1Þ and n even. (ii) n odd. In this case we let FðxÞ :¼ xn for all x 2 R; by 5.2.3(a), F is continuous on R. We leave it to the reader to show that F is strictly increasing on R and that FðRÞ ¼ R. (See Figure 5.6.6.) It follows from the Continuous Inverse Theorem 5.6.5 that the function G that is inverse to FðxÞ ¼ xn for x 2 R, is strictly increasing and continuous on R. We usually write pffiffiffi GðxÞ ¼ x1=n or GðxÞ ¼ n x for x 2 R; n odd; and call x1=n the nth root of x 2 R. The function G is called the nth root function (n odd). (See Figure 5.6.7.) Here we have 1=n n ðxn Þ1=n ¼ x and x ¼x for all x 2 R and n odd. Figure 5.6.6 Graph of F(x) ¼ xn (x 2 R, n odd) Figure 5.6.7 Graph of G(x) ¼ x1=n (x 2 R, n odd) C05 12/08/2010 14:20:15 Page 159 5.6 MONOTONE AND INVERSE FUNCTIONS 159 Rational Powers Now that the nth root functions have been defined for n 2 N, it is easy to define rational powers. 5.6.6 Definition (i) If m, n 2 N and x 0, we define xm=n :¼ x1=n m (ii) If m, n 2 N and x > 0, we define xm=n :¼ x1=n . m . Hence we have defined xr when r is a rational number and x > 0. The graphs of x 7! xr depend on whether r > 1, r ¼ 1, 0 < r < 1, r ¼ 0, or r < 0. (See Figure 5.6.8.) Since a rational number r 2 Q can be written in the form r ¼ m=n with m 2 Z, n 2 N, in many ways, it should be shown that Definition 5.6.6is not ambiguous. That is, if r ¼ m=n ¼ p=q m p with m, p 2 Z and n, q 2 N and if x > 0, then x1=n ¼ x1=q . We leave it as an exercise to the reader to establish this relation. 5.6.7 Theorem If m 2 Z, n 2 N, and x > 0, then xm=n ¼ ðxm Þ1=n . m Proof. If x > 0 and m, n 2 Z, then (xm)n ¼ xmn ¼ (xn)m. Now let y :¼ xm=n ¼ x1=n > 0 n m m n ¼ x1=n ¼ xm . Therefore it follows that y ¼ ðxm Þ1=n . so that yn ¼ x1=n Q.E.D. The reader should also show, as an exercise, that if x > 0 and r, s 2 Q , then xr xs ¼ xrþs ¼ xs xr Figure 5.6.8 and ðxr Þs ¼ xrs ¼ ðxs Þr: Graphs of x ! xr ðx 0Þ C05 12/08/2010 14:20:16 160 Page 160 CHAPTER 5 CONTINUOUS FUNCTIONS Exercises for Section 5.6 1. If I :¼ [a, b] is an interval and f : I ! R is an increasing function, then the point a [respectively, b] is an absolute minimum [respectively, maximum] point for f on I. If f is strictly increasing, then a is the only absolute minimum point for f on I. 2. If f and g are increasing functions on an interval I R, show that f þ g is an increasing function on I. If f is also strictly increasing on I, then f þ g is strictly increasing on I. 3. Show that both f (x) :¼ x and g(x) :¼ x 1 are strictly increasing on I :¼ [0, 1], but that their product f g is not increasing on I. 4. Show that if f and g are positive increasing functions on an interval I, then their product fg is increasing on I. 5. Show that if I :¼ [a, b] and f : I ! R is increasing on I, then f is continuous at a if and only if f (a) ¼ inf{ f (x) : x 2 (a, b]}. 6. Let I R be an interval and let f : I ! R be increasing on I. Suppose that c 2 I is not an endpoint of I. Show that f is continuous at c if and only if there exists a sequence (xn) in I such that xn < c for n ¼ 1, 3, 5, . . . ; xn > c for n ¼ 2, 4, 6, . . . ; and such that c ¼ lim(xn) and f (c) ¼ lim ( f (xn)). 7. Let I R be an interval and let f : I ! R be increasing on I. If c is not an endpoint of I, show that the jump jf (c) of f at c is given by inf{ f (y) f (x) : x < c < y, x, y 2 I}. 8. Let f, g be strictly increasing on an interval I R and let f (x) > g(x) for all x 2 I. If y 2 f ðI Þ \ gðI Þ, show that f 1 ðyÞ < g1 ðyÞ. [Hint: First interpret this statement geometrically.] 9. Let I :¼ [0, 1] and let f : I ! R be defined by f (x) :¼ x for x rational, and f (x) :¼ 1 x for x irrational. Show that f is injective on I and that f ( f (x)) ¼ x for all x 2 I. (Hence f is its own inverse function!) Show that f is continuous only at the point x ¼ 12. 10. Let I :¼ [a, b] and let f : I ! R be continuous on I. If f has an absolute maximum [respectively, minimum] at an interior point c of I, show that f is not injective on I. 11. Let f (x) :¼ x for x 2 [0, 1], and f (x) :¼ 1 þ x for x 2 (1, 2]. Show that f and f 1 are strictly increasing. Are f and f 1 continuous at every point? 12. Let f : ½0; 1 ! R be a continuous function that does not take on any of its values twice and with f (0) < f (1). Show that f is strictly increasing on [0, 1]. 13. Let h : ½0; 1 ! R be a function that takes on each of its values exactly twice. Show that h cannot be continuous at every point. [Hint: If c1 < c2 are the points where h attains its supremum, show that c1 ¼ 0, c2 ¼ 1. Now examine the points where h attains its infimum.] m p 14. Let x 2 R, x > 0. Show that if m, p 2 Z, n, q 2 N, and mq ¼ np, then x1=n ¼ x1=q . 15. If x 2 R, x > 0, and if r, s 2 Q , show that x r x s ¼ x rþs ¼ x s x r and ðx r Þs ¼ x rs ¼ ðx s Þr . C06 12/09/2010 14:53:55 Page 161 CHAPTER 6 DIFFERENTIATION Prior to the seventeenth century, a curve was generally described as a locus of points satisfying some geometric condition, and tangent lines were obtained through geometric construction. This viewpoint changed dramatically with the creation of analytic geometry in the 1630s by Rene Descartes (1596–1650) and Pierre de Fermat (1601–1665). In this new setting geometric problems were recast in terms of algebraic expressions, and new classes of curves were defined by algebraic rather than geometric conditions. The concept of derivative evolved in this new context. The problem of finding tangent lines and the seemingly unrelated problem of finding maximum or minimum values were first seen to have a connection by Fermat in the 1630s. And the relation between tangent lines to curves and the velocity of a moving particle was discovered in the late 1660s by Isaac Newton. Newton’s theory of ‘‘fluxions,’’ which was based on an intuitive idea of limit, would be familiar to any modern student of differential calculus once some changes in terminology and notation were made. But the vital observation, made by Newton and, independently, by Gottfried Leibniz in the 1680s, was that areas under curves could be calculated by reversing the differentiation process. This exciting technique, one that solved previously difficult area problems with ease, sparked enormous interest among the mathematicians of the era and led to a coherent theory that became known as the differential and integral calculus. Isaac Newton Isaac Newton (1642–1727) was born in Woolsthorpe, in Lincolnshire, England, on Christmas Day; his father, a farmer, had died three months earlier. His mother remarried when he was three years old and he was sent to live with his grandmother. He returned to his mother at age eleven, only to be sent to boarding school in Grantham the next year. Fortunately, a perceptive teacher noticed his mathematical talent and, in 1661, Newton entered Trinity College at Cambridge University, where he studied with Isaac Barrow. When the bubonic plague struck in 1665–1666, leaving dead nearly 70,000 persons in London, the university closed and Newton spent two years back in Woolsthorpe. It was during this period that he formulated his basic ideas concerning optics, gravitation, and his method of ‘‘fluxions,’’ later called ‘‘calculus.’’ He returned to Cambridge in 1667 and was appointed Lucasian Professor in 1669. His theories of universal gravitation and planetary motion were published to world acclaim in 1687 under the title Philosophies Naturalis Principia Mathematica. However, he neglected to publish his method of inverse tangents for finding areas and other work in calculus, and this led to a controversy over priority with Leibniz. Following an illness, he retired from Cambridge University and in 1696 was appointed Warden of the British mint. However, he maintained contact with advances in science and mathematics and served as President of the Royal Society from 1703 until his death in 1727. At his funeral, Newton was eulogized as ‘‘the greatest genius that ever existed.’’ His place of burial in Westminster Abbey is a popular tourist site. 161 C06 12/09/2010 14:53:55 162 Page 162 CHAPTER 6 DIFFERENTIATION In this chapter we will develop the theory of differentiation. Integration theory, including the fundamental theorem that relates differentiation and integration, will be the subject of the next chapter. We will assume that the reader is already familiar with the geometrical and physical interpretations of the derivative of a function as described in introductory calculus courses. Consequently, we will concentrate on the mathematical aspects of the derivative and not go into its applications in geometry, physics, economics, and so on. The first section is devoted to a presentation of the basic results concerning the differentiation of functions. In Section 6.2 we discuss the fundamental Mean Value Theorem and some of its applications. In Section 6.3 the important L’Hospital Rules are presented for the calculation of certain types of ‘‘indeterminate’’ limits. In Section 6.4 we give a brief discussion of Taylor’s Theorem and a few of its applications—for example, to convex functions and to Newton’s Method for the location of roots. Section 6.1 The Derivative In this section we will present some of the elementary properties of the derivative. We begin with the definition of the derivative of a function. 6.1.1 Definition Let I R be an interval, let f : I ! R, and let c 2 I. We say that a real number L is the derivative of f at c if given any e > 0 there exists dðeÞ > 0 such that if x 2 I satisfies 0 < jx cj < dðeÞ, then f ðxÞ f ðcÞ < e: ð1Þ L xc In this case we say that f is differentiable at c, and we write f 0 ðcÞ for L. In other words, the derivative of f at c is given by the limit f ðxÞ f ðcÞ x!c xc f 0 ðcÞ ¼ lim ð2Þ provided this limit exists. (We allow the possibility that c may be the endpoint of the interval.) Note It is possible to define the derivative of a function having a domain more general than an interval (since the point c need only be an element of the domain and also a cluster point of the domain) but the significance of the concept is most naturally apparent for functions defined on intervals. Consequently we shall limit our attention to such functions. Whenever the derivative of f : I ! R exists at a point c 2 I, its value is denoted by f 0 ðcÞ. In this way we obtain a function f 0 whose domain is a subset of the domain of f. In working with the function f 0 , it is convenient to regard it also as a function of x. For example, if f ðxÞ :¼ x2 for x 2 R, then at any c in R we have f ðxÞ f ðcÞ x2 c2 ¼ lim ¼ lim ðx þ cÞ ¼ 2c: x!c x!c x c x!c xc f 0 ðcÞ ¼ lim Thus, in this case, the function f 0 is defined on all of R and f 0 ðxÞ ¼ 2x for x 2 R . C06 12/09/2010 14:53:55 Page 163 6.1 THE DERIVATIVE 163 We now show that continuity of f at a point c is a necessary (but not sufficient) condition for the existence of the derivative at c. 6.1.2 Theorem If f : I ! R has a derivative at c 2 I, then f is continuous at c. Proof. For all x 2 I; x 6¼ c, we have f ðxÞ f ðcÞ ¼ f ðxÞ f ðcÞ ðx cÞ: xc Since f 0 ðcÞ exists, we may apply Theorem 4.2.4 concerning the limit of a product to conclude that f ðxÞ f ðcÞ lim ð f ðxÞ f ðcÞÞ ¼ lim lim ðx cÞ x!c x!c x!c xc ¼ f 0 ðcÞ 0 ¼ 0: Therefore, lim f ðxÞ ¼ f ðcÞ so that f is continuous at c. x!c Q.E.D. The continuity of f : I ! R at a point does not assure the existence of the derivative at that point. For example, if f ðxÞ :¼ jxj for x 2 R, then for x 6¼ 0 we have ð f ðxÞ f ð0ÞÞ=ðx 0Þ ¼ jxj=x, which is equal to 1 if x > 0, and equal to 1 if x < 0. Thus the limit at 0 does not exist [see Example 4.1.10(b)], and therefore the function is not differentiable at 0. Hence, continuity at a point c is not a sufficient condition for the derivative to exist at c. Remark By taking simple algebraic combinations of functions of the form x 7! jx cj, it is not difficult to construct continuous functions that do not have a derivative at a finite (or even a countable) number of points. In 1872, Karl Weierstrass astounded the mathematical world by giving an example of a function that is continuous at every point but whose derivative does not exist anywhere. Such a function defied geometric intuition about curves and tangent lines, and consequently spurred much deeper investigations into the concepts of real analysis. It can be shown that the function f defined by the series f ðxÞ :¼ 1 X 1 n n cos ð3 xÞ 2 n¼0 has the stated property. A very interesting historical discussion of this and other examples of continuous, nondifferentiable functions is given in Kline, pp. 955–966, and also in Hawkins, pp. 44–46. A detailed proof for a slightly different example can be found in Appendix E. There are a number of basic properties of the derivative that are very useful in the calculation of the derivatives of various combinations of functions. We now provide the justification of some of these properties, which will be familiar to the reader from earlier courses. 6.1.3 Theorem Let I R be an interval, let c 2 I, and let f : I ! R and g : I ! R be functions that are differentiable at c. Then: (a) If a 2 R, then the function a f is differentiable at c, and ð3Þ ða f Þ0 ðcÞ ¼ a f 0 ðcÞ: C06 12/09/2010 14:53:56 164 Page 164 CHAPTER 6 DIFFERENTIATION (b) The function f þ g is differentiable at c, and ð f þ gÞ0 ðcÞ ¼ f 0 ðcÞ þ g0 ðcÞ: ð4Þ (c) (Product Rule) The function fg is differentiable at c, and ð f gÞ0 ðcÞ ¼ f 0 ðcÞgðcÞ þ f ðcÞg0 ðcÞ: ð5Þ (d) (Quotient Rule) If gðcÞ 6¼ 0, then the function f=g is differentiable at c, and 0 f f 0 ðcÞgðcÞ f ðcÞg0 ðcÞ ðcÞ ¼ : ð6Þ g ð gð c Þ Þ 2 Proof. We shall prove (c) and (d), leaving (a) and (b) as exercises for the reader. (c) Let p :¼ f g; then for x 2 I; x 6¼ c, we have pðxÞ pðcÞ f ðxÞgðxÞ f ðcÞgðcÞ ¼ xc xc f ðxÞgðxÞ f ðcÞgðxÞ þ f ðcÞgðxÞ f ðcÞgðcÞ ¼ xc f ðxÞ f ðcÞ gðxÞ gðcÞ ¼ gðxÞ þ f ðcÞ : xc xc Since g is continuous at c, by Theorem 6.1.2, then lim gðxÞ ¼ gðcÞ. Since f and g are x!c differentiable at c, we deduce from Theorem 4.2.4 on properties of limits that pðxÞ pðcÞ ¼ f 0 ðcÞgðcÞ þ f ðcÞg0 ðcÞ: xc Hence p :¼ f g is differentiable at c and (5) holds. (d) Let q :¼ f =g. Since g is differentiable at c, it is continuous at that point (by Theorem 6.1.2). Therefore, since gðcÞ 6¼ 0, we know from Theorem 4.2.9 that there exists an interval J I with c 2 J such that gðxÞ 6¼ 0 for all x 2 J. For x 2 J; x 6¼ c, we have lim x!c qðxÞ qðcÞ f ðxÞ=gðxÞ f ðcÞ=gðcÞ f ðxÞgðcÞ f ðcÞgðxÞ ¼ ¼ xc xc gðxÞgðcÞðx cÞ f ðxÞgðcÞ f ðcÞgðcÞ þ f ðcÞgðcÞ f ðcÞgðxÞ ¼ gðxÞgðcÞðx cÞ 1 f ðxÞ f ðcÞ gðxÞ gðcÞ gðcÞ f ðcÞ ¼ : gðxÞgðcÞ xc xc Using the continuity of g at c and the differentiability of f and g at c, we get qðxÞ qðcÞ f 0 ðcÞgðcÞ f ðcÞg0 ðcÞ ¼ q0 ðcÞ ¼ lim : x!c xc ðgðcÞÞ2 Thus, q ¼ f =g is differentiable at c and equation (6) holds. Q.E.D. Mathematical Induction may be used to obtain the following extensions of the differentiation rules. 6.1.4 Corollary If f 1 ; f 2 ; . . . ; f n are functions on an interval I to R that are differentiable at c 2 I, then: (a) The function f 1 þ f 2 þ þ f n is differentiable at c and ð7Þ ð f 1 þ f 2 þ þ f n Þ0 ðcÞ ¼ f 01 ðcÞ þ f 02 ðcÞ þ þ f 0n ðcÞ: C06 12/09/2010 14:53:56 Page 165 6.1 THE DERIVATIVE 165 (b) The function f 1 f 2 f n is differentiable at c, and ð8Þ ð f 1 f 2 f n Þ0 ðcÞ ¼ f 01 ðcÞ f 2 ðcÞ f n ðcÞ þ f 1 ðcÞ f 02 ðcÞ f n ðcÞ þ þ f 1 ðcÞ f 2 ðcÞ f 0n ðcÞ: An important special case of the extended product rule (8) occurs if the functions are equal, that is, f 1 ¼ f 2 ¼ ¼ f n ¼ f . Then (8) becomes ð f n Þ0 ðcÞ ¼ nð f ðcÞÞn1 f 0 ðcÞ: ð9Þ In particular, if we take f ðxÞ :¼ x, then we find the derivative of gðxÞ :¼ xn to be g0 ðxÞ ¼ nxn1; n 2 N. The formula is extended to include negative integers by applying the Quotient Rule 6.1.3(d). Notation If I R is an interval and f : I ! R, we have introduced the notation f 0 to denote the function whose domain is a subset of I and whose value at a point c is the derivative f 0 ðcÞ of f at c. There are other notations that are sometimes used for f 0; for example, one sometimes writes Df for f 0. Thus one can write formulas (4) and (5) in the form: Dð f þ gÞ ¼ Df þ Dg; Dð f gÞ ¼ ðDf Þ g þ f ðDgÞ: When x is the ‘‘independent variable,’’ it is common practice in elementary courses to write df =dx for f 0. Thus formula (5) is sometimes written in the form d df dg ð f ðxÞgðxÞÞ ¼ ðxÞ gðxÞ þ f ðxÞ ðxÞ : dx dx dx This last notation, due to Leibniz, has certain advantages. However, it also has certain disadvantages and must be used with some care. The Chain Rule We now turn to the theorem on the differentiation of composite functions known as the ‘‘Chain Rule.’’ It provides a formula for finding the derivative of a composite function g f in terms of the derivatives of g and f. We first establish the following theorem concerning the derivative of a function at a point that gives us a very nice method for proving the Chain Rule. It will also be used to derive the formula for differentiating inverse functions. 6.1.5 Caratheodory’s Theorem Let f be defined on an interval I containing the point c. Then f is differentiable at c if and only if there exists a function w on I that is continuous at c and satisfies ð10Þ f ðxÞ f ðcÞ ¼ wðxÞðx cÞ f or x 2 I: 0 In this case, we have wðcÞ ¼ f ðcÞ. Proof. ð)Þ If f 0 ðcÞ exists, we can define w by 8 < f ðxÞ f ðcÞ for wðxÞ :¼ xc : 0 f ð cÞ for x 6¼ c; x 2 I; x ¼ c: The continuity of w follows from the fact that lim wðxÞ ¼ f 0 ðcÞ. If x ¼ c, then both sides of x!c (10) equal 0, while if x 6¼ c, then multiplication of wðxÞ by x c gives (10) for all other x 2 I. C06 12/09/2010 14:53:57 166 Page 166 CHAPTER 6 DIFFERENTIATION ð(Þ Now assume that a function w that is continuous at c and satisfying (10) exists. If we divide (10) by x c 6¼ 0, then the continuity of w implies that wðcÞ ¼ lim wðxÞ ¼ lim x!c x!c f ðxÞ f ðcÞ xc exists. Therefore f is differentiable at c and f 0 ðcÞ ¼ wðcÞ. Q.E.D. To illustrate Caratheodory’s Theorem, we consider the function f defined by f ðxÞ :¼ x3 for x 2 R. For c 2 R, we see from the factorization x3 c3 ¼ x2 þ cx þ c2 ðx cÞ that wðxÞ :¼ x2 þ cx þ c2 satisfies the conditions of the theorem. Therefore, we conclude that f is differentiable at c 2 R and that f 0 ðcÞ ¼ wðcÞ ¼ 3c2 . We will now establish the Chain Rule. If f is differentiable at c and g is differentiable at f (c), then the Chain Rule states that the derivative of the composite function g f at c is the product ðg f Þ0 ðcÞ ¼ g0 ð f ðcÞÞ f 0 ðcÞ. Note that this can be written as ð g f Þ 0 ¼ ð g0 f Þ f 0 : One approach to the Chain Rule is the observation that the difference quotient can be written, when f ðxÞ 6¼ f ðcÞ, as the product gð f ðxÞÞ gð f ðcÞÞ gð f ðxÞÞ gð f ðcÞÞ f ðxÞ f ðcÞ ¼ : xc f ð x Þ f ð cÞ xc This suggests the correct limiting value. Unfortunately, the first factor in the product on the right is undefined if the denominator f ðxÞ f ðcÞ equals 0 for values of x near c, and this presents a problem. However, the use of Caratheodory’s Theorem neatly avoids this difficulty. 6.1.6 Chain Rule Let I, J be intervals in R, let g : I ! R and f : J ! R be functions such that f ðJ Þ I, and let c 2 J. If f is differentiable at c and if g is differentiable at f (c), then the composite function g f is differentiable at c and ð11Þ ðg f Þ0 ðcÞ ¼ g0 ð f ðcÞÞ f 0 ðcÞ: Proof. Since f 0 ðcÞ exists, Caratheodory’s Theorem 6.1.5 implies that there exists a function w on J such that w is continuous at c and f ðxÞ f ðcÞ ¼ wðxÞðx cÞ for x 2 J, and where wðcÞ ¼ f 0 ðcÞ. Also, since g0 ð f ðcÞÞ exists, there is a function c defined on I such that c is continuous at d :¼ f ðcÞ and gðyÞ gðd Þ ¼ cðyÞðy d Þ for y 2 I, where cðd Þ ¼ g0 ðd Þ. Substitution of y ¼ f ðxÞ and d ¼ f ðcÞ then produces gð f ðxÞÞ gð f ðcÞÞ ¼ cð f ðxÞÞð f ðxÞ f ðcÞÞ ¼ ½ðc f ðxÞÞ wðxÞðx cÞ for all x 2 J such that f ðxÞ 2 I. Since the function ðc f Þ w is continuous at c and its value at c is g0 ð f ðcÞÞ f 0 ðcÞ, Caratheodory’s Theorem gives (11). Q.E.D. If g is differentiable on I, if f is differentiable on J, and if f ðJ Þ I, then it follows from the Chain Rule that ðg f Þ0 ¼ ðg0 f Þ f 0 , which can also be written in the form Dðg f Þ ¼ ðDg f Þ Df . C06 12/09/2010 14:53:57 Page 167 6.1 THE DERIVATIVE 167 6.1.7 Examples (a) If f : I ! R is differentiable on I and gðyÞ :¼ yn for y 2 R and n 2 N, then since g0 ðyÞ ¼ nyn1 , it follows from the Chain Rule 6.1.6 that ðg f Þ0 ðxÞ ¼ g0 ð f ðxÞÞ f 0 ðxÞ x 2 I: for Therefore we have ð f n Þ0 ðxÞ ¼ nð f ðxÞÞn1 f 0 ðxÞ for all x 2 I as was seen in (9). (b) Suppose that f : I ! R is differentiable on I and that f ðxÞ 6¼ 0 and f 0 ðxÞ 6¼ 0 for x 2 I. If hðyÞ :¼ 1=y for y 6¼ 0, then it is an exercise to show that h0 ðyÞ ¼ 1=y2 for y 2 R; y 6¼ 0. Therefore we have 0 1 f 0 ðxÞ ðxÞ ¼ ðh f Þ0 ðxÞ ¼ h0 ð f ðxÞÞ f 0 ðxÞ ¼ for x 2 I: f ð f ðxÞÞ2 (c) The absolute value function gðxÞ :¼ jxj is differentiable at all x 6¼ 0 and has derivative g0 ðxÞ ¼ sgnðxÞ for x 6¼ 0. (The signum function is defined in Example 4.1.10(b).) Though sgn is defined everywhere, it is not equal to g0 at x ¼ 0 since g0 ð0Þ does not exist. Now if f is a differentiable function, then the Chain Rule implies that the function g f ¼ j f j is also differentiable at all points x where f ðxÞ 6¼ 0, and its derivative is given by j f j0 ðxÞ ¼ sgnð f ðxÞÞ f 0 ðxÞ ¼ f 0 ðxÞ if f ðxÞ > 0; f 0 ðxÞ if f ðxÞ < 0: If f is differentiable at a point c with f ðcÞ ¼ 0, then it is an exercise to show that j f j is differentiable at c if and only if f 0 ðcÞ ¼ 0. (See Exercise 7.) 2 For example, 2 if f ðxÞ :¼ x 0 1 for x 22R, then the derivative of its absolute value j f jðxÞ ¼ x 1 is equal to j f j ðxÞ ¼ sgnðx 1Þ ð2xÞ for x 6¼ 1; 1. See Figure 6.1.1 for a graph of j f j. Figure 6.1.1 The function j f jðxÞ ¼ x2 1. (d) It will be proved later that if SðxÞ :¼ sin x and CðxÞ :¼ cos x for all x 2 R, then S0 ðxÞ ¼ cos x ¼ C ðxÞ and C 0 ðxÞ ¼ sin x ¼ SðxÞ for all x 2 R. If we use these facts together with the definitions tan x :¼ sin x 1 ; sec x :¼ ; cos x cos x C06 12/09/2010 14:53:58 168 Page 168 CHAPTER 6 DIFFERENTIATION for x 6¼ ð2k þ 1Þp=2; k 2 Z , and apply the Quotient Rule 6.1.3(d), we obtain ðcos xÞðcos xÞ ðsin xÞðsin xÞ ¼ ðsec xÞ2 ; ðcos xÞ2 0 1ðsin xÞ sin x ¼ ¼ ðsec xÞðtan xÞ D sec x ¼ 2 ðcos xÞ ðcos xÞ2 D tan x ¼ for x 6¼ ð2k þ 1Þp=2; k 2 Z . Similarly, since cot x :¼ cos x ; sin x csc x :¼ 1 sin x for x 6¼ kp; k 2 Z , then we obtain D cot x ¼ ðcsc xÞ2 D csc x ¼ ðcsc xÞðcot xÞ and for x 6¼ kp; k 2 Z . (e) Suppose that f is defined by f ðxÞ :¼ x2 sin ð1=xÞ 0 for for x 6¼ 0; x ¼ 0: If we use the fact that D sin x ¼ cos x for all x 2 R and apply the Product Rule 6.1.3(c) and the Chain Rule 6.1.6, we obtain (why?) f 0 ðxÞ ¼ 2x sin ð1=xÞ cos ð1=xÞ for x 6¼ 0: If x ¼ 0, none of the calculational rules may be applied. (Why?) Consequently, the derivative of f at x ¼ 0 must be found by applying the definition of derivative. We find that f ðxÞ f ð0Þ x2 sin ð1=xÞ ¼ lim ¼ lim x sin ð1=xÞ ¼ 0: x!0 x!0 x!0 x0 x f 0 ð0Þ ¼ lim Hence, the derivative f 0 of f exists at all x 2 R. However, the function f 0 does not have a limit at x ¼ 0 (why?), and consequently f 0 is discontinuous at x ¼ 0. Thus, a function f that & is differentiable at every point of R need not have a continuous derivative f 0. Inverse Functions We will now relate the derivative of a function to the derivative of its inverse function, when this inverse function exists. We will limit our attention to a continuous strictly monotone function and use the Continuous Inverse Theorem 5.6.5 to ensure the existence of a continuous inverse function. If f is a continuous strictly monotone function on an interval I, then its inverse function g ¼ f 1 is defined on the interval J :¼ f ðI Þ and satisfies the relation g f ðxÞ ¼ x for x 2 I: If c 2 I and d :¼ f ðcÞ, and if we knew that both f 0 ðcÞ and g0 ðd Þ exist, then we could differentiate both sides of the equation and apply the Chain Rule to the left side to get g0 f ðcÞ f 0 ðcÞ ¼ 1. Thus, if f 0 ðcÞ 6¼ 0, we would obtain g0 ð d Þ ¼ 1 : f 0 ðcÞ C06 12/09/2010 14:53:58 Page 169 6.1 THE DERIVATIVE 169 However, it is necessary to deduce the differentiability of the inverse function g from the assumed differentiability of f before such a calculation can be performed. This is nicely accomplished by using Caratheodory’s Theorem. 6.1.8 Theorem Let I be an interval in R and let f : I ! R be strictly monotone and continuous on I. Let J :¼ f ðI Þ and let g : J ! R be the strictly monotone and continuous function inverse to f. If f is differentiable at c 2 I and f 0 ðcÞ 6¼ 0, then g is differentiable at d :¼ f ðcÞ and g0 ð d Þ ¼ ð12Þ 1 1 ¼ : f 0 ðcÞ f 0 ðgðd ÞÞ Proof. Given c 2 R, we obtain from Caratheodory’s Theorem 6.1.5 a function w on I with properties that w is continuous at c; f ðxÞ f ðcÞ ¼ wðxÞðx cÞ for x 2 I, and wðcÞ ¼ f 0 ðcÞ. Since wðcÞ 6¼ 0 by hypothesis, there exists a neighborhood V :¼ ðc d; c þ dÞ such that wðxÞ 6¼ 0 for all x 2 V \ I. (See Theorem 4.2.9.) If U :¼ f ðV \ I Þ, then the inverse function g satisfies f ðgðyÞÞ ¼ y for all y 2 U, so that y d ¼ f ðgðyÞÞ f ðcÞ ¼ wðgðyÞÞ ðgðyÞ gðd ÞÞ: Since wðgðyÞÞ 6¼ 0 for y 2 U, we can divide to get gðyÞ gðd Þ ¼ 1 ðy d Þ: w ð gð y Þ Þ Since the function 1=ðw gÞ is continuous at d, we apply Theorem 6.1.5 to conclude that g0 ðd Þ exists and g0 ðd Þ ¼ 1=wðgðd ÞÞ ¼ 1=wðcÞ ¼ 1=f 0 ðcÞ: Q.E.D. Note The hypothesis, made in Theorem 6.1.8, that f 0 ðcÞ 6¼ 0 is essential. In fact, if f 0 ðcÞ ¼ 0, then the inverse function g is never differentiable at d ¼ f ðcÞ, since the assumed existence of g0 ðd Þ would lead to 1 ¼ f 0 ðcÞg0 ðd Þ ¼ 0, which is impossible. The function f ðxÞ :¼ x3 with c ¼ 0 is such an example. 6.1.9 Theorem Let I be an interval and let f : I ! R be strictly monotone on I. Let J :¼ f ðI Þ and let g : J ! R be the function inverse to f. If f is differentiable on I and f 0 ðxÞ 6¼ 0 for x 2 I, then g is differentiable on J and g0 ¼ ð13Þ 1 : f0 g Proof. If f is differentiable on I, then Theorem 6.1.2 implies that f is continuous on I, and by the Continuous Inverse Theorem 5.6.5, the inverse function g is continuous on J. Equation (13) now follows from Theorem 6.1.8. Q.E.D. Remark If f and g are the functions of Theorem 6.1.9, and if x 2 I and y 2 J are related by y ¼ f ðxÞ and x ¼ gðyÞ, then equation (13) can be written in the form g0 ð y Þ ¼ 1 ; y 2 J; ð f gÞ ð y Þ 0 or ðg0 f ÞðxÞ ¼ 1 ; x 2 I: f ðxÞ 0 It can also be written in the form g0 ðyÞ ¼ 1=f 0 ðxÞ, provided that it is kept in mind that x and y are related by y ¼ f ðxÞ and x ¼ gðyÞ. C06 12/09/2010 14:53:59 170 Page 170 CHAPTER 6 DIFFERENTIATION 6.1.10 Examples (a) The function f : R ! R defined by f ðxÞ :¼ x5 þ 4x þ 3 is continuous and strictly monotone increasing (since it is the sum of two strictly increasing functions). Moreover, f 0 ðxÞ ¼ 5x4 þ 4 is never zero. Therefore, by Theorem 6.1.8, the inverse function g ¼ f 1 is differentiable at every point. If we take c ¼ 1, then since f ð1Þ ¼ 8, we obtain g0 ð8Þ ¼ g0 ð f ð1ÞÞ ¼ 1=f 0 ð1Þ ¼ 1=9. (b) Let n 2 N be even, let I :¼ ½0; 1Þ, and let f ðxÞ :¼ xn for x 2 I. It was seen at the end of Section 5.6 that f is strictly increasing and continuous on I, so that its inverse function gðyÞ :¼ y1=n for y 2 J :¼ ½0; 1Þ is also strictly increasing and continuous on J. Moreover, we have f 0 ðxÞ ¼ nx n1 for all x 2 I. Hence it follows that if y > 0, then g0 ðyÞ exists and g0 ðyÞ ¼ 1 1 1 ¼ ¼ ðn1Þ=n : n1 f ðgðyÞÞ nðgðyÞÞ ny 0 Hence we deduce that 1 g0 ðyÞ ¼ yð1=nÞ1 for y > 0: n However, g is not differentiable at 0. (For a graph of f and g, see Figures 5.6.4 and 5.6.5.) (c) Let n 2 N, n 6¼ 1, be odd, let F ðxÞ :¼ xn for x 2 R, and let GðyÞ :¼ y1=n be its inverse function defined for all y 2 R. As in part (b) we find that G is differentiable for y 6¼ 0 and that G0 ðyÞ ¼ ð1=nÞyð1=nÞ1 for y 6¼ 0. However, G is not differentiable at 0, even though G is differentiable for all y 6¼ 0. (For a graph of F and G, see Figures 5.6.6 and 5.6.7.) (d) Let r :¼ m=n be a positive rational number, let I :¼ ½0; 1Þ, and let RðxÞ :¼ xr for x 2 I. (Recall Definition 5.6.6.) Then R is the composition of the functions f ðxÞ :¼ xm and gðxÞ :¼ x1=n ; x 2 I. That is, RðxÞ ¼ f ðgðxÞÞ for x 2 I. If we apply the Chain Rule 6.1.6 and the results of (b) [or (c), depending on whether n is even or odd], then we obtain m1 1 ð1=nÞ1 R0 ðxÞ ¼ f 0 ðgðxÞÞg0 ðxÞ ¼ m x1=n x n m ðm=nÞ1 ¼ x ¼ rxr1 n for all x > 0. If r > 1, then it is an exercise to show that the derivative also exists at x ¼ 0 and R0 ð0Þ ¼ 0. (For a graph of R see Figure 5.6.8.) (e) The sine function is strictly increasing on the interval I :¼ ½p=2; p=2; therefore its inverse function, which we will denote by Arcsin, exists on J :¼ ½1; 1. That is, if x 2 ½p=2; p=2 and y 2 ½1; 1 then y ¼ sin x if and only if Arcsin y ¼ x. It was asserted (without proof) in Example 6.1.7(d) that sin is differentiable on I and that D sin x ¼ cos x for x 2 I. Since cos x 6¼ 0 for x in ðp=2; p=2Þ it follows from Theorem 6.1.8 that 1 1 ¼ D sin x cos x 1 1 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 y2 1 ðsin xÞ D Arcsin y ¼ for all y 2 ð1; 1Þ. The derivative of Arcsin does not exist at the points 1 and 1. Exercises for Section 6.1 1. Use the definition to find the derivative of each of the following functions: (a) f ðxÞ :¼ xp3ffiffiffifor x 2 R; (b) gðxÞ :¼ 1=x pfor ffiffiffi x 2 R; x 6¼ 0; (c) hðxÞ :¼ x for x > 0; (d) kðxÞ :¼ 1= x for x > 0: & C06 12/09/2010 14:54:0 Page 171 6.1 THE DERIVATIVE 171 2. Show that f ðxÞ :¼ x1=3 ; x 2 R, is not differentiable at x ¼ 0. 3. Prove Theorem 6.1.3(a), (b). 4. Let f : R ! R be defined by f ðxÞ :¼ x2 for x rational, f ðxÞ :¼ 0 for x irrational. Show that f is differentiable at x ¼ 0, and find f 0 ð0Þ. 5. Differentiate and simplify: x (a) f ðxÞ :¼ ; 1 þ x2 k m for m; k 2 N; (c) hðxÞ :¼ sin x pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 2x þ x2 ; (b) gðxÞ :¼ (d) kðxÞ :¼ tan ðx2 Þ for jxj < pffiffiffiffiffiffiffiffi p=2: 6. Let n 2 N and let f : R ! R be defined by f ðxÞ :¼ xn for x 0 and f ðxÞ :¼ 0 for x < 0. For which values of n is f 0 continuous at 0? For which values of n is f 0 differentiable at 0? 7. Suppose that f : R ! R is differentiable at c and that f ðcÞ ¼ 0. Show that gðxÞ :¼ j f ðxÞj is differentiable at c if and only if f 0 ðcÞ ¼ 0. 8. Determine where each of the following functions from R to R is differentiable and find the derivative: (a) f ðxÞ :¼ jxj þ jx þ 1j; (b) gðxÞ :¼ 2x þ jxj; (c) hðxÞ :¼ xjxj; (d) kðxÞ :¼ jsin xj: 9. Prove that if f : R ! R is an even function [that is, f ðxÞ ¼ f ðxÞ for all x 2 R] and has a derivative at every point, then the derivative f 0 is an odd function [that is, f 0 ðxÞ ¼ f 0 ðxÞ for all x 2 R]. Also prove that if g : R ! R is a differentiable odd function, then g0 is an even function. 10. Let g : R ! R be defined by gðxÞ :¼ x2 sin ð1=x2 Þ for x 6¼ 0, and gð0Þ :¼ 0. Show that g is differentiable for all x 2 R. Also show that the derivative g0 is not bounded on the interval ½1; 1. 11. Assume that there exists a function L : ð0; 1Þ ! R such that L0 ðxÞ ¼ 1=x for x > 0. Calculate the derivatives of the following functions: 3 (a) f ðxÞ :¼ Lð2x þ 3Þ for x > 0; (b) gðxÞ :¼ ðLðx2 ÞÞ for x > 0; (c) hðxÞ :¼ LðaxÞ for a > 0; x > 0; (d) kðxÞ :¼ LðLðxÞÞ when LðxÞ > 0; x > 0: 12. If r > 0 is a rational number, let f : R ! R be defined by f ðxÞ :¼ xr sin ð1=xÞ for x 6¼ 0, and f ð0Þ :¼ 0. Determine those values of r for which f 0 ð0Þ exists. 13. If f : R ! R is differentiable at c 2 R, show that f 0 ðcÞ ¼ limðnf f ðc þ 1=nÞ f ðcÞgÞ: However, show by example that the existence of the limit of this sequence does not imply the existence of f 0 ðcÞ. 14. Given that the function hðxÞ :¼ x3 þ 2x þ 1 for x 2 R has an inverse h1 on R, find the value of 0 h1 ðyÞ at the points corresponding to x ¼ 0; 1; 1. 15. Given that the restriction of the cosine function cos to I :¼ ½0; p is strictly decreasing and that cos 0 ¼ 1; cos p ¼ 1, let J :¼ ½1; 1, and let Arccos: J ! R be the function inverse to the restriction of cos to I. Show that Arccos is differentiable on ð1; 1Þ and D Acrccos y ¼ 1=2 ð1Þ=ð1 y2 Þ for y 2 ð1; 1Þ. Show that Arccos is not differentiable at 1 and 1. 16. Given that the restriction of the tangent function tan to I :¼ ðp=2; p=2Þ is strictly increasing and that tan ðI Þ ¼ R, let Arctan: R ! R be the function inverse to the restriction of tan to I. Show that Arctan is differentiable on R and that D ArctanðyÞ ¼ ð1 þ y2 Þ1 for y 2 R. 17. Let f : I ! R be differentiable at c 2 I. Establish the Straddle Lemma: Given e > 0 there exists dðeÞ > 0 such that if u, v 2 I satisfy c dðeÞ < u c v < c þ dðeÞ, then we have j f ðvÞ f ðuÞ ðv uÞ f 0 ðcÞj eðv uÞ. [Hint: The dðeÞ is given by Definition 6.1.1. Subtract and add the term f ðcÞ cf 0 ðcÞ on the left side and use the Triangle Inequality.] C06 12/09/2010 14:54:0 172 Page 172 CHAPTER 6 DIFFERENTIATION Section 6.2 The Mean Value Theorem The Mean Value Theorem, which relates the values of a function to values of its derivative, is one of the most useful results in real analysis. In this section we will establish this important theorem and sample some of its many consequences. We begin by looking at the relationship between the relative extrema of a function and the values of its derivative. Recall that the function f : I ! R is said to have a relative maximum [respectively, relative minimum] at c 2 I if there exists a neighborhood V :¼ V d ðcÞ of c such that f ðxÞ f ðcÞ [respectively, f ðcÞ f ðxÞ] for all x in V \ I. We say that f has a relative extremum at c 2 I if it has either a relative maximum or a relative minimum at c. The next result provides the theoretical justification for the familiar process of finding points at which f has relative extrema by examining the zeros of the derivative. However, it must be realized that this procedure applies only to interior points of the interval. For example, if f ðxÞ :¼ x on the interval I :¼ ½0; 1, then the endpoint x ¼ 0 yields the unique relative minimum and the endpoint x ¼ 1 yields the unique maximum of f on I, but neither point is a zero of the derivative of f. 6.2.1 Interior Extremum Theorem Let c be an interior point of the interval I at which f : I ! R has a relative extremum. If the derivative of f at c exists, then f 0 ðcÞ ¼ 0. Proof. We will prove the result only for the case that f has a relative maximum at c; the proof for the case of a relative minimum is similar. If f 0 ðcÞ > 0, then by Theorem 4.2.9 there exists a neighborhood V I of c such that f ðxÞ f ðcÞ > 0 for xc If x 2 V and x > c, then we have x 2 V; x 6¼ c: f ðxÞ f ðcÞ > 0: xc But this contradicts the hypothesis that f has a relative maximum at c. Thus we cannot have f 0 ðcÞ > 0. Similarly (how?), we cannot have f 0 ðcÞ < 0. Therefore we must have f 0 ðcÞ ¼ 0. Q.E.D. f ð xÞ f ð c Þ ¼ ð x c Þ 6.2.2 Corollary Let f : I ! R be continuous on an interval I and suppose that f has a relative extremum at an interior point c of I. Then either the derivative of f at c does not exist, or it is equal to zero. We note that if f ðxÞ :¼ jxj on I :¼ ½1; 1, then f has an interior minimum at x ¼ 0; however, the derivative of f fails to exist at x ¼ 0. 6.2.3 Rolle’s Theorem Suppose that f is continuous on a closed interval I :¼ ½a; b, that the derivative f 0 exists at every point of the open interval (a, b), and that f ðaÞ ¼ f ðbÞ ¼ 0. Then there exists at least one point c in (a, b) such that f 0 ðcÞ ¼ 0. Proof. If f vanishes identically on I, then any c in (a, b) will satisfy the conclusion of the theorem. Hence we suppose that f does not vanish identically; replacing f by f if necessary, we may suppose that f assumes some positive values. By the MaximumMinimum Theorem 5.3.4, the function f attains the value supf f ðxÞ : x 2 I g > 0 at some point c in I. Since f ðaÞ ¼ f ðbÞ ¼ 0, the point c must lie in (a, b); therefore f 0 ðcÞ exists. C06 12/09/2010 14:54:1 Page 173 6.2 THE MEAN VALUE THEOREM Figure 6.2.1 173 Rolle’s Theorem Since f has a relative maximum at c, we conclude from the Interior Extremum Theorem 6.2.1 that f 0 ðcÞ ¼ 0. (See Figure 6.2.1.) Q.E.D. As a consequence of Rolle’s Theorem, we obtain the fundamental Mean Value Theorem. 6.2.4 Mean Value Theorem Suppose that f is continuous on a closed interval I :¼ ½a; b, and that f has a derivative in the open interval (a, b). Then there exists at least one point c in (a, b) such that f ðbÞ f ðaÞ ¼ f 0 ðcÞðb aÞ: Proof. Consider the function w defined on I by wðxÞ :¼ f ðxÞ f ðaÞ f ð bÞ f ð aÞ ðx aÞ: ba [The function w is simply the difference of f and the function whose graph is the line segment joining the points (a, f (a)) and (b, f (b)); see Figure 6.2.2.] The hypotheses of Figure 6.2.2 The Mean Value Theorem C06 12/09/2010 14:54:2 174 Page 174 CHAPTER 6 DIFFERENTIATION Rolle’s Theorem are satisfied by w since w is continuous on [a, b], differentiable on (a, b), and wðaÞ ¼ wðbÞ ¼ 0. Therefore, there exists a point c in (a, b) such that 0 ¼ w 0 ð cÞ ¼ f 0 ð cÞ f ðbÞ f ðaÞ : ba Hence, f ðbÞ f ðaÞ ¼ f 0 ðcÞðb aÞ. Q.E.D. Remark The geometric view of the Mean Value Theorem is that there is some point on the curve y ¼ f ðxÞ at which the tangent line is parallel to the line segment through the points (a, f (a)) and (b, f (b)). Thus it is easy to remember the statement of the Mean Value Theorem by drawing appropriate diagrams. While this should not be discouraged, it tends to suggest that its importance is geometrical in nature, which is quite misleading. In fact the Mean Value Theorem is a wolf in sheep’s clothing and is the Fundamental Theorem of Differential Calculus. In the remainder of this section, we will present some of the consequences of this result. Other applications will be given later. The Mean Value Theorem permits one to draw conclusions about the nature of a function f from information about its derivative f 0 . The following results are obtained in this manner. 6.2.5 Theorem Suppose that f is continuous on the closed interval I :¼ ½a; b, that f is differentiable on the open interval (a, b), and that f 0 ðxÞ ¼ 0 for x 2 ða; bÞ. Then f is constant on I. Proof. We will show that f ðxÞ ¼ f ðaÞ for all x 2 I. Indeed, if x 2 I, x > a, is given, we apply the Mean Value Theorem to f on the closed interval ½a, x. We obtain a point c (depending on x) between a and x such that f ðxÞ f ðaÞ ¼ f 0 ðcÞðx aÞ. Since f 0 ðcÞ ¼ 0 (by hypothesis), we deduce that f ðxÞ f ðaÞ ¼ 0. Hence, f ðxÞ ¼ f ðaÞ for any x 2 I. Q.E.D. 6.2.6 Corollary Suppose that f and g are continuous on I :¼ ½a; b, that they are differentiable on (a, b), and that f 0 ðxÞ ¼ g0 ðxÞ for all x 2 ða; bÞ. Then there exists a constant C such that f ¼ g þ C on I. Recall that a function f : I ! R is said to be increasing on the interval I if whenever x1 ; x2 in I satisfy x1 < x2, then f ðx1 Þ f ðx2 Þ. Also recall that f is decreasing on I if the function f is increasing on I. 6.2.7 Theorem Let f : I ! R be differentiable on the interval I. Then: (a) f is increasing on I if and only if f 0 ðxÞ 0 for all x 2 I. (b) f is decreasing on I if and only if f 0 ðxÞ 0 for all x 2 I. Proof. (a) Suppose that f 0 ðxÞ 0 for all x 2 I. If x1 ; x2 in I satisfy x1 < x2, then we apply the Mean Value Theorem to f on the closed interval J :¼ ½x1 ; x2 to obtain a point c in ðx1 ; x2 Þ such that f ðx2 Þ f ðx1 Þ ¼ f 0 ðcÞðx2 x1 Þ: C06 12/09/2010 14:54:3 Page 175 6.2 THE MEAN VALUE THEOREM 175 Since f 0 ðcÞ 0 and x2 x1 > 0, it follows that f ðx2 Þ f ðx1 Þ 0. (Why?) Hence, f ðx1 Þ f ðx2 Þ and, since x1 < x2 are arbitrary points in I, we conclude that f is increasing on I. For the converse assertion, we suppose that f is differentiable and increasing on I. Thus, for any point x 6¼ c in I, we have ð f ðxÞ f ðcÞÞ=ðx cÞ 0. (Why?) Hence, by Theorem 4.2.6 we conclude that f 0 ðcÞ ¼ lim x!c f ðxÞ f ðcÞ 0: xc (b) The proof of part (b) is similar and will be omitted. Q.E.D. A function f is said to be strictly increasing on an interval I if for any points x1 ; x2 in I such that x1 < x2 , we have f ðx1 Þ < f ðx2 Þ. An argument along the same lines of the proof of Theorem 6.2.7 can be made to show that a function having a strictly positive derivative on an interval is strictly increasing there. (See Exercise 13.) However, the converse assertion is not true, since a strictly increasing differentiable function may have a derivative that vanishes at certain points. For example, the function f : R ! R defined by f ðxÞ :¼ x3 is strictly increasing on R, but f 0 ð0Þ ¼ 0. The situation for strictly decreasing functions is similar. Remark It is reasonable to define a function to be increasing at a point if there is a neighborhood of the point on which the function is increasing. One might suppose that, if the derivative is strictly positive at a point, then the function is increasing at this point. However, this supposition is false; indeed, the differentiable function defined by gðxÞ :¼ x þ 2x2 sin ð1=xÞ 0 if if x 6¼ 0; x ¼ 0; is such that g0 ð0Þ ¼ 1, yet it can be shown that g is not increasing in any neighborhood of x ¼ 0. (See Exercise 10.) We next obtain a sufficient condition for a function to have a relative extremum at an interior point of an interval. 6.2.8 First Derivative Test for Extrema Let f be continuous on the interval I :¼ ½a; b and let c be an interior point of I. Assume that f is differentiable on (a, c) and (c, b) . Then: (a) and (b) and If there is a neighborhood ðc d; c þ dÞ I such that f 0 ðxÞ 0 for c d < x < c f 0 ðxÞ 0 for c < x < c þ d, then f has a relative maximum at c. If there is a neighborhood ðc d; c þ dÞ I such that f 0 ðxÞ 0 for c d < x < c f 0 ðxÞ 0 for c < x < c þ d, then f has a relative minimum at c. Proof. (a) If x 2 ðc d; cÞ, then it follows from the Mean Value Theorem that there exists a point cx 2 ðx; cÞ such that f ðcÞ f ðxÞ ¼ ðc xÞ f 0 ðcx Þ. Since f 0 ðcx Þ 0 we infer that f ðxÞ f ðcÞ for x 2 ðc d; cÞ. Similarly, it follows (how?) that f ðxÞ f ðcÞ for x 2 ðc; c þ dÞ. Therefore f ðxÞ f ðcÞ for all x 2 ðc d; c þ dÞ so that f has a relative maximum at c. (b) The proof is similar. Q.E.D. Remark The converse of the First Derivative Test 6.2.8 is not true. For example, there exists a differentiable function f : R ! R with absolute minimum at x ¼ 0 but such that C06 12/09/2010 14:54:3 176 Page 176 CHAPTER 6 DIFFERENTIATION f 0 takes on both positive and negative values on both sides of (and arbitrarily close to) x ¼ 0. (See Exercise 9.) Further Applications of the Mean Value Theorem We will continue giving other types of applications of the Mean Value Theorem; in doing so we will draw more freely than before on the past experience of the reader and his or her knowledge concerning the derivatives of certain well-known functions. 6.2.9 Examples (a) Rolle’s Theorem can be used for the location of roots of a function. For, if a function g can be identified as the derivative of a function f, then between any two roots of f there is at least one root of g. For example, let gðxÞ :¼ cos x, then g is known to be the derivative of f ðxÞ :¼ sin x. Hence, between any two roots of sin x there is at least one root of cos x. On the other hand, g0 ðxÞ ¼ sin x ¼ f ðxÞ, so another application of Rolle’s Theorem tells us that between any two roots of cos there is at least one root of sin. Therefore, we conclude that the roots of sin and cos interlace each other. This conclusion is probably not news to the reader; however, the same type of argument can be applied to the Bessel functions Jn of order n ¼ 0; 1; 2; . . . by using the relations 0 ½xn J n ðxÞ0 ¼ xn J n1 ðxÞ; ½xn1 J n ðxÞ ¼ xn J nþ1 ðxÞ for x > 0: The details of this argument should be supplied by the reader. (b) We can apply the Mean Value Theorem for approximatepcalculations and to obtain ffiffiffiffiffiffiffiffi error estimates. For example, suppose it is desired to evaluate 105 . We employ the Mean pffiffiffi Value Theorem with f ðxÞ :¼ x; a ¼ 100; b ¼ 105, to obtain pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi 5 105 100 ¼ pffiffiffi 2 c pffiffiffi pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi for some number c with 100 < c < 105. Since 10 < c < 105 < 121 ¼ 11, we can assert that pffiffiffiffiffiffiffiffi 5 5 < 105 10 < ; 2ð11Þ 2ð10Þ pffiffiffiffiffiffiffiffi whence it follows that 10:2272 < 105p <ffiffiffi 10:2500. estimate may not be as sharp as pffiffiffiffiffiffiffiffi This pffiffiffiffiffiffiffi ffi desired. It is clear that the estimate c < 105 < 121 was wasteful can be pffiffiffiffiffiffiffiffi pffiffiand ffi improved by making use of our conclusion that 105 < 10:2500. Thus, c < 10:2500 and we easily determine that pffiffiffiffiffiffiffiffi 5 < 105 10: 2ð10:2500Þ pffiffiffiffiffiffiffiffi Our improved estimate is 10:2439 < 105 < 10:2500. 0:2439 < & Inequalities One very important use of the Mean Value Theorem is to obtain certain inequalities. Whenever information concerning the range of the derivative of a function is available, this information can be used to deduce certain properties of the function itself. The following examples illustrate the valuable role that the Mean Value Theorem plays in this respect. C06 12/09/2010 14:54:3 Page 177 6.2 THE MEAN VALUE THEOREM 177 6.2.10 Examples (a) The exponential function f ðxÞ :¼ ex has the derivative f 0 ðxÞ ¼ ex for all x 2 R. Thus f 0 ðxÞ > 1 for x > 0, and f 0 ðxÞ < 1 for x < 0. From these relationships, we will derive the inequality ð1Þ ex 1 þ x for x 2 R; with equality occurring if and only if x ¼ 0. If x ¼ 0, we have equality with both sides equal to 1. If x > 0, we apply the Mean Value Theorem to the function f on the interval [0, x]. Then for some c with 0 < c < x we have ex e0 ¼ ec ðx 0Þ: Since e0 ¼ 1 and ec > 1, this becomes ex 1 > x so that we have ex > 1 þ x for x > 0. A similar argument establishes the same strict inequality for x < 0. Thus the inequality (1) holds for all x, and equality occurs only if x ¼ 0. (b) The function gðxÞ :¼ sin x has the derivative g0 ðxÞ ¼ cos x for all x 2 R. On the basis of the fact that 1 cos x 1 for all x 2 R, we will show that ð2Þ x sin x x for all x 0: Indeed, if we apply the Mean Value Theorem to g on the interval [0, x], where x > 0, we obtain sin x sin 0 ¼ ðcos cÞðx 0Þ for some c between 0 and x. Since sin 0 ¼ 0 and 1 cos c 1, we have x sin x x. Since equality holds at x ¼ 0, the inequality (2) is established. (c) (Bernoulli’s inequality) If a > 1, then ð3Þ ð1 þ xÞa 1 þ ax for all x > 1; with equality if and only if x ¼ 0. This inequality was established earlier, in Example 2.1.13(c), for positive integer values of a by using Mathematical Induction. We now derive the more general version by employing the Mean Value Theorem. If hðxÞ :¼ ð1 þ xÞa then h0 ðxÞ ¼ að1 þ xÞa1 for all x > 1. [For rational a this derivative was established in Example 6.1.10(c). The extension to irrational will be discussed in Section 8.3.] If x > 0, we infer from the Mean Value Theorem applied to h on the interval [0, x] that there exists c with 0 < c < x such that hðxÞ hð0Þ ¼ h0 ðcÞðx 0Þ. Thus, we have ð1 þ xÞa 1 ¼ að1 þ cÞa1 x: Since c > 0 and a 1 > 0, it follows that ð1 þ cÞa1 > 1 and hence that ð1 þ xÞa > 1 þ ax. If 1 < x < 0, a similar use of the Mean Value Theorem on the interval [x, 0] leads to the same strict inequality. Since the case x ¼ 0 results in equality, we conclude that (3) is valid for all x > 1 with equality if and only if x ¼ 0. (d) Let a be a real number satisfying 0 < a < 1 and let gðxÞ ¼ ax xa for x 0. Then g0 ðxÞ ¼ að1 xa1 Þ, so that g0 ðxÞ < 0 for 0 < x < 1 and g0 ðxÞ > 0 for x > 1. Consequently, if x 0, then gðxÞ gð1Þ and gðxÞ ¼ gð1Þ if and only if x ¼ 1. Therefore, if x 0 and 0 < a < 1, then we have xa a x þ ð1 aÞ: C06 12/09/2010 14:54:4 178 Page 178 CHAPTER 6 DIFFERENTIATION If a > 0 and b > 0 and if we let x ¼ a=b and multiply by b, we obtain the inequality aa b1a aa þ ð1 aÞb; & where equality holds if and only if a ¼ b. The Intermediate Value Property of Derivatives We conclude this section with an interesting result, often referred to as Darboux’s Theorem. It states that if a function f is differentiable at every point of an interval I, then the function f 0 has the Intermediate Value Property. This means that if f 0 takes on values A and B, then it also takes on all values between A and B. The reader will recognize this property as one of the important consequences of continuity as established in Theorem 5.3.7. It is remarkable that derivatives, which need not be continuous functions, also possess this property. 6.2.11 Lemma Let I R be an interval, let f : I ! R, let c 2 I, and assume that f has a derivative at c. Then: (a) If f 0 ðcÞ > 0, then there is a number d > 0 such that f ðxÞ > f ðcÞ for x 2 I such that c < x < c þ d. (b) If f 0 ðcÞ < 0, then there is a number d > 0 such that f ðxÞ > f ðcÞ for x 2 I such that c d < x < c. Proof. (a) Since f ðxÞ f ðcÞ ¼ f 0 ðcÞ > 0; x!c xc it follows from Theorem 4.2.9 that there is a number d > 0 such that if x 2 I and 0 < jx cj < d, then lim f ðxÞ f ðcÞ > 0: xc If x 2 I also satisfies x > c, then we have f ðxÞ f ðcÞ > 0: xc Hence, if x 2 I and c < x < c þ d, then f ðxÞ > f ðcÞ. The proof of (b) is similar. f ðxÞ f ðcÞ ¼ ðx cÞ Q.E.D. 6.2.12 Darboux’s Theorem If f is differentiable on I ¼ ½a; b and if k is a number between f 0 ðaÞ and f 0 ðbÞ, then there is at least one point c in (a, b) such that f 0 ðcÞ ¼ k. Proof. Suppose that f 0 ðaÞ < k < f 0 ðbÞ. We define g on I by gðxÞ :¼ kx f ðxÞ for x 2 I. Since g is continuous, it attains a maximum value on I. Since g0 ðaÞ ¼ k f 0 ðaÞ > 0, it follows from Lemma 6.2.11(a) that the maximum of g does not occur at x ¼ a. Similarly, since g0 ðbÞ ¼ k f 0 ðbÞ < 0, it follows from Lemma 6.2.11(b) that the maximum does not occur at x ¼ b. Therefore, g attains its maximum at some c in (a, b). Then from Theorem 6.2.1 we have 0 ¼ g0 ðcÞ ¼ k f 0 ðcÞ. Hence, f 0 ðcÞ ¼ k. Q.E.D. 6.2.13 Example The function g: ½1; 1 ! R 8 for < 1 gðxÞ :¼ 0 for : 1 for defined by 0 < x 1; x ¼ 0; 1 x < 0; C06 12/09/2010 14:54:4 Page 179 6.2 THE MEAN VALUE THEOREM 179 (which is a restriction of the signum function) clearly fails to satisfy the intermediate value property on the interval ½1; 1. Therefore, by Darboux’s Theorem, there does not exist a function f such that f 0 ðxÞ ¼ gðxÞ for all x 2 ½1; 1. In other words, g is not the derivative & on ½1; 1 of any function. Exercises for Section 6.2 1. For each of the following functions on R to R, find points of relative extrema, the intervals on which the function is increasing, and those on which it is decreasing: (a) f ðxÞ :¼ x2 3x þ 5; (b) gðxÞ :¼ 3x 4x2 ; (c) hðxÞ :¼ x3 3x 4; (d) kðxÞ :¼ x4 þ 2x2 4: 2. Find the points of relative extrema, the intervals on which the following functions are increasing, and those on which they are decreasing: (a) f ðxÞ :¼ x þ 1=x for x 6¼ 0; (b) gðxÞ :¼ x=ðx2 þ 1Þ for x 2 R; pffiffiffiffiffiffiffiffiffiffiffi pffiffiffi (c) hðxÞ :¼ x 2 x þ 2 for x > 0; (d) kðxÞ :¼ 2x þ 1=x2 for x 6¼ 0: 3. Find the points extrema of the following functions on the specified domain: of relative (b) gðxÞ :¼ 1 ðx 1Þ2=3 for 0 x 2; (a) f ðxÞ :¼ x2 1 for 4 x 4; 2 (d) kðxÞ :¼ xðx 8Þ1=3 for 0 x 9: (c) hðxÞ :¼ xx 12 for 2 x 3; 4. Let a1 ; a2 ; . . . ; an be real numbers and let f be defined on R by n X ðai xÞ2 for x 2 R: f ðxÞ :¼ i¼1 Find the unique point of relative minimum for f. 5. Let a > b > 0 and let n 2 N satisfy n 2. Prove that a1=n b1=n < ða bÞ1=n . [Hint: Show that f ðxÞ :¼ x1=n ðx 1Þ1=n is decreasing for x 1, and evaluate f at 1 and a=b.] 6. Use the Mean Value Theorem to prove that jsin x sin yj jx yj for all x, y in R. 7. Use the Mean Value Theorem to prove that ðx 1Þ=x < ln x < x 1 for x > 1. [Hint: Use the fact that D ln x ¼ 1=x for x > 0.] 8. Let f : ½a; b ! R be continuous on [a, b] and differentiable in (a, b). Show that if lim f 0 ðxÞ ¼ A, x!a then f 0 ðaÞ exists and equals A. [Hint: Use the definition of f 0 ðaÞ and the Mean Value Theorem.] 9. Let f : R ! R be defined by f ðxÞ :¼ 2x4 þ x4 sinð1=xÞ for x 6¼ 0 and f ð0Þ :¼ 0. Show that f has an absolute minimum at x ¼ 0, but that its derivative has both positive and negative values in every neighborhood of 0. 10. Let g : R ! R be defined by gðxÞ :¼ x þ 2x2 sinð1=xÞ for x 6¼ 0 and gð0Þ :¼ 0. Show that g0 ð0Þ ¼ 1, but in every neighborhood of 0 the derivative g0 ðxÞ takes on both positive and negative values. Thus g is not monotonic in any neighborhood of 0. 11. Give an example of a uniformly continuous function on [0,1] that is differentiable on (0, 1) but whose derivative is not bounded on (0, 1). 12. If hðxÞ :¼ 0 for x < 0 and hðxÞ :¼ 1 for x 0, prove there does not exist a function f : R ! R such that f 0 ðxÞ ¼ hðxÞ for all x 2 R. Give examples of two functions, not differing by a constant, whose derivatives equal h(x) for all x 6¼ 0. 13. Let I be an interval and let f : I ! R be differentiable on I. Show that if f 0 is positive on I, then f is strictly increasing on I. 14. Let I be an interval and let f : I ! R be differentiable on I. Show that if the derivative f 0 is never 0 on I, then either f 0 ðxÞ > 0 for all x 2 I or f 0 ðxÞ < 0 for all x 2 I. 15. Let I be an interval. Prove that if f is differentiable on I and if the derivative f 0 is bounded on I, then f satisfies a Lipschitz condition on I. (See Definition 5.4.4.) C06 12/09/2010 14:54:5 180 Page 180 CHAPTER 6 DIFFERENTIATION 16. Let f : ½0; 1Þ ! R be differentiable on (0, 1) and assume that f 0 ðxÞ ! b as x ! 1. (a) Show that for any h > 0, we have lim ð f ðx þ hÞ f ðxÞÞ=h ¼ b. x!1 (b) Show that if f ðxÞ ! a as x ! 1, then b ¼ 0. (c) Show that lim ð f ðxÞ=xÞ ¼ b. x!1 17. Let f, g be differentiable on R and suppose that f ð0Þ ¼ gð0Þ and f 0 ðxÞ g0 ðxÞ for all x 0. Show that f ðxÞ gðxÞ for all x 0. 18. Let I :¼ ½a; b and let f : I ! R be differentiable at c 2 I. Show that for every e > 0 there exists d > 0 such that if 0 < jx yj < d and a x c y b, then f ðxÞ f ðyÞ 0 < e: f ð c Þ xy 19. A differentiable function f : I ! R is said to be uniformly differentiable on I :¼ ½a; b if for every e > 0 there exists d > 0 such that if 0 < jx yj < d and x; y 2 I, then f ðx Þ f ðy Þ 0 < e: f ð x Þ xy 0 Show that if f is uniformly differentiable on I, then f is continuous on I. 20. Suppose that f : ½0; 2 ! R is continuous on [0, 2] and differentiable on (0, 2), and that f ð0Þ ¼ 0; f ð1Þ ¼ 1; f ð2Þ ¼ 1. (a) Show that there exists c1 2 ð0; 1Þ such that f 0 ðc1 Þ ¼ 1. (b) Show that there exists c2 2 ð1; 2Þ such that f 0 ðc2 Þ ¼ 0. (c) Show that there exists c 2 ð0; 2Þ such that f 0 ðcÞ ¼ 1=3. Section 6.3 L’Hospital’s Rules In this section we will discuss limit theorems that involve cases that cannot be determined by previous limit theorems. For example, if f (x) and g(x) both approach 0 as x approaches a, then the quotient f ðxÞ=gðxÞ may or may not have a limit at a and it is said to have the indeterminate form 0/0. The limit theorem for this case is due to Johann Bernoulli and first appeared in the 1696 book published by L’Hospital. Johann Bernoulli Johann Bernoulli (1667–1748) was born in Basel, Switzerland. Johann worked for a year in his father’s spice business, but he was not a success. He enrolled in Basel University to study medicine, but his brother Jacob, twelve years older and a Professor of Mathematics, led him into mathematics. Together, they studied the papers of Leibniz on the new subject of calculus. Johann received his doctorate at Basel University and joined the faculty at Groningen in Holland, but upon Jacob’s death in 1705, he returned to Basel and was awarded Jacob’s chair in mathematics. Because of his many advances in the subject, Johann is regarded as one of the founders of calculus. While in Paris in 1692, Johann met the Marquis Guillame Francois de L’Hospital and agreed to a financial arrangement under which he would teach the new calculus to L’Hospital, giving L’Hospital the right to use Bernoulli’s lessons as he pleased. This was subsequently continued through a series of letters. In 1696, the first book on differential calculus, L’Analyse des Infiniment Petits, was published by L’Hospital. Though L’Hospital’s name was not on the title page, his portrait was on the frontispiece and the preface states ‘‘I am indebted to the clarifications of the brothers Bernoulli, especially the younger.’’ The book contains a theorem on limits later known as L’Hospital’s Rule although it was in fact discovered by Johann Bernoulli. In 1922, manuscripts were discovered that confirmed the book consisted mainly of Bernoulli’s lessons. And in 1955, the L’Hospital–Bernoulli correspondence was published in Germany. C06 12/09/2010 14:54:5 Page 181 6.3 L’HOSPITAL’S RULES 181 The initial theorem was refined and extended, and the various results are collectively referred to as L’Hospital’s (or L’H^ opital’s) Rules. In this section we establish the most basic of these results and indicate how others can be derived. Indeterminate Forms In the preceding chapters we have often been concerned with methods of evaluating limits. It was shown in Theorem 4.2.4(b) that if A :¼ lim f ðxÞ and B :¼ lim gðxÞ, and if B 6¼ 0, then x!c x!c f ðxÞ A ¼ : x!c gðxÞ B lim However, if B ¼ 0, then no conclusion was deduced. It will be seen in Exercise 2 that if B ¼ 0 and A 6¼ 0, then the limit is infinite (when it exists). The case A ¼ 0, B ¼ 0 has not been covered previously. In this case, the limit of the quotient f=g is said to be ‘‘indeterminate.’’ We will see that in this case the limit may not exist or may be any real value, depending on the particular functions f and g. The symbolism 0=0 is used to refer to this situation. For example, if a is any real number, and if we define f ðxÞ :¼ ax and gðxÞ :¼ x, then f ð xÞ ax lim ¼ lim ¼ lim a ¼ a: x!0 gðxÞ x!0 x x!0 Thus the indeterminate form 0=0 can lead to any real number a as a limit. Other indeterminate forms are represented by the symbols 1=1; 0 1; 00 ; 11 ; 10 ; and 1 1. These notations correspond to the indicated limiting behavior and juxtaposition of the functions f and g. Our attention will be focused on the indeterminate forms 0=0 and 1=1. The other indeterminate cases are usually reduced to the form 0=0 or 1=1 by taking logarithms, exponentials, or algebraic manipulations. A Preliminary Result To show that the use of differentiation in this context is a natural and not surprising development, we first establish an elementary result that is based simply on the definition of the derivative. 6.3.1 Theorem Let f and g be defined on [a, b], let f ðaÞ ¼ gðaÞ ¼ 0, and let gðxÞ 6¼ 0 for a < x < b. If f and g are differentiable at a and if g0 ðaÞ 6¼ 0, then the limit of f=g at a exists and is equal to f 0 ðaÞ=g0 ðaÞ. Thus f ðxÞ f 0 ðaÞ ¼ 0 : lim x!aþ gðxÞ g ðaÞ Proof. Since f ðaÞ ¼ gðaÞ ¼ 0, we can write the quotient f ðxÞ=gðxÞ for a < x < b as follows: f ðxÞ f ðaÞ f ðxÞ f ðxÞ f ðaÞ xa ¼ ¼ gðxÞ gðxÞ gðaÞ gðxÞ gðaÞ : xa Applying Theorem 4.2.4(b), we obtain f ðxÞ f ðaÞ lim f ðxÞ x!aþ f 0 ð aÞ xa lim ¼ ¼ 0 : Q.E.D. x!aþ gðxÞ gðxÞ gðaÞ g ðaÞ lim x!aþ xa C06 12/09/2010 14:54:5 182 Page 182 CHAPTER 6 DIFFERENTIATION Warning The hypothesis that f ðaÞ ¼ gðaÞ ¼ 0 is essential here. For example, if f ðxÞ :¼ x þ 17 and gðxÞ :¼ 2x þ 3 for x 2 R, then lim x!0 f ðxÞ 17 ¼ ; gðxÞ 3 while f 0 ð0Þ 1 ¼ : g0 ð 0Þ 2 The preceding result enables us to deal with limits such as x2 þ x 2 0 þ 1 1 ¼ ¼ : x!0 sin 2x 2 cos 0 2 lim To handle limits where f and g are not differentiable at the point a, we need a more general version of the Mean Value Theorem due to Cauchy. 6.3.2 Cauchy Mean Value Theorem Let f and g be continuous on [a, b] and differentiable on (a, b), and assume that g0 ðxÞ 6¼ 0 for all x in (a, b). Then there exists c in (a, b) such that f ðbÞ f ðaÞ f 0 ðcÞ ¼ : gðbÞ gðaÞ g0 ðcÞ Proof. As in the proof of the Mean Value Theorem, we introduce a function to which Rolle’s Theorem will apply. First we note that since g0 ðxÞ 6¼ 0 for all x in (a, b), it follows from Rolle’s Theorem that gðaÞ 6¼ gðbÞ. For x in [a, b], we now define hðxÞ :¼ f ð bÞ f ð aÞ ðgðxÞ gðaÞÞ ð f ðxÞ f ðaÞÞ: gð bÞ gð aÞ Then h is continuous on [a, b], differentiable on (a, b), and hðaÞ ¼ hðbÞ ¼ 0. Therefore, it follows from Rolle’s Theorem 6.2.3 that there exists a point c in (a, b) such that 0 ¼ h0 ð c Þ ¼ f ð bÞ f ð aÞ 0 g ðcÞ f 0 ðcÞ: gð bÞ gð aÞ Since g0 ðcÞ 6¼ 0, we obtain the desired result by dividing by g0 ðcÞ. Q.E.D. Remarks The preceding theorem has a geometric interpretation that is similar to that of the Mean Value Theorem 6.2.4. The functions f and g can be viewed as determining a curve in the plane by means of the parametric equations x ¼ f ðtÞ; y ¼ gðtÞ where a t b. Then the conclusion of the theorem is that there exists a point ð f ðcÞ; gðcÞÞ on the curve for some c in (a, b) such that the slope g0 ðcÞ=f 0 ðcÞ of the line tangent to the curve at that point is equal to the slope of the line segment joining the endpoints of the curve. Note that if gðxÞ ¼ x, then the Cauchy Mean Value Theorem reduces to the Mean Value Theorem 6.2.4. L’Hospital’s Rule, I We will now establish the first of L’Hospital’s Rules. For convenience, we will consider right-hand limits at a point a; left-hand limits, and two-sided limits are treated in exactly the same way. In fact, the theorem even allows the possibility that a ¼ 1. The reader should observe that, in contrast with Theorem 6.3.1, the following result does not assume the differentiability of the functions at the point a. The result asserts that the limiting behavior of f ðxÞ=gðxÞ as x ! aþ is the same as the limiting behavior of f 0 ðxÞ=g0 ðxÞ as C06 12/09/2010 14:54:6 Page 183 6.3 L’HOSPITAL’S RULES 183 x ! aþ, including the case where this limit is infinite. An important hypothesis here is that both f and g approach 0 as x ! aþ. 6.3.3 L’Hospital’s Rule, I Let 1 a < b 1 and let f, g be differentiable on (a, b) such that g0 ðxÞ 6¼ 0 for all x 2 ða; bÞ . Suppose that ð1Þ lim f ðxÞ ¼ 0 ¼ lim gðxÞ: x!aþ x!aþ f 0 ðxÞ f ðxÞ ¼ L: ¼ L 2 R; then lim x!aþ g0 ðxÞ x!aþ gðxÞ (a) If lim f 0 ðxÞ f ðxÞ ¼ L: ¼ L 2 f1; 1g; then lim x!aþ g0 ðxÞ x!aþ gðxÞ (b) If lim Proof. If a < a < b < b, then Rolle’s Theorem implies that gðbÞ 6¼ gðaÞ. Further, by the Cauchy Mean Value Theorem 6.3.2, there exists u 2 ða; bÞ such that f ðbÞ f ðaÞ f 0 ðuÞ ¼ : gðbÞ gðaÞ g0 ðuÞ ð2Þ Case (a): If L 2 R and if e > 0 is given, there exists c 2 ða; bÞ such that Le < f 0 ðuÞ <Lþe g0 ð uÞ u 2 ða; cÞ; for whence it follows from (2) that ð3Þ Le < f ðbÞ f ðaÞ <Lþe gðbÞ gðaÞ for a < a < b c: If we take the limit in (3) as a ! aþ, we have Le f ðbÞ Lþe gð b Þ for b 2 ða; c: Since e > 0 is arbitrary, the assertion follows. Case (b): If L ¼ þ1 and if M > 0 is given, there exists c 2 ða; bÞ such that f 0 ð uÞ >M g0 ð uÞ u 2 ða; cÞ; for whence it follows from (2) that ð4Þ f ð bÞ f ð a Þ >M gð b Þ gð a Þ for a < a < b < c: If we take the limit in (4) as a ! aþ, we have f ðbÞ M gð b Þ for Since M > 0 is arbitrary, the assertion follows. If L ¼ 1, the argument is similar. b 2 ða; cÞ: Q.E.D. The corresponding theorem for left-hand limits is readily proved in the same manner. The result for two-sided limits then follows immediately if both one-sided limits exist and C06 12/09/2010 14:54:6 184 Page 184 CHAPTER 6 DIFFERENTIATION are equal. In the examples that follow, we will apply the appropriate version of L’Hospital’s Rule as needed. 6.3.4 Examples (a) We have pffiffiffi sin x cos x pffiffiffi ¼ lim 2 x cos x ¼ 0: lim pffiffiffi ¼ lim x!0þ x!0þ x!0þ x 1=ð2 xÞ Observe that the denominator is not differentiable pffiffiffiat x ¼ 0 so that Theorem 6.3.1 cannot be applied. However f ðxÞ :¼ sin x and gðxÞ :¼ x are differentiable on (0, 1) and both approach 0 as x ! 0þ. Moreover, g0 ðxÞ 6¼ 0 on (0, 1), so that 6.3.3 is applicable. 1 cos x sin x : (b) We have lim ¼ lim x!0 x!0 2x x2 The quotient in the second limit is again indeterminate in the form 0=0. However, the hypotheses are again satisfied so that a second application of L’Hospital’s Rule is permissible. Hence, we obtain 1 cos x sin x cos x 1 lim ¼ lim ¼ : ¼ lim 2 x!0 x!0 2x x!0 2 x 2 ex 1 ex ¼ lim ¼ 1. x!0 x!0 1 x Similarly, two applications of L’Hospital’s Rule give us x e 1x ex 1 ex 1 ¼ lim ¼ : lim ¼ lim x!0 x!0 2x x!0 2 x2 2 ln x ð1=xÞ ¼ 1. (d) We have lim ¼ lim x!1 x 1 x!1 1 (c) We have lim & L’Hospital’s Rule, II This rule is very similar to the first one, except that it treats the case where the denominator becomes infinite as x ! aþ. Again we will consider only right-hand limits, but it is possible that a ¼ 1. Left-hand limits and two-sided limits are handled similarly. 6.3.5 L’Hospital’s Rule, II Let 1 a < b 1 and let f, g be differentiable on (a, b) such that g0 ðxÞ 6¼ 0 for all x 2 ða; bÞ . Suppose that ð5Þ lim gðxÞ ¼ x!aþ 1: f 0 ðxÞ f ðxÞ ¼ L. ¼ L 2 R, then lim x!aþ g0 ðxÞ x!aþ gðxÞ (a) If lim f 0 ðxÞ f ðxÞ ¼ L. ¼ L 2 f1; 1g, then lim 0 x!aþ g ðxÞ x!aþ gðxÞ (b) If lim Proof. We will suppose that (5) holds with limit 1. As before, we have gðbÞ 6¼ gðaÞ for a; b 2 ða; bÞ; a < b. Further, equation (2) in the proof of 6.3.3 holds for some u 2 ða; bÞ. Case (a): If L 2 R with L > 0 and e > 0 is given, there is c 2 ða; bÞ such that (3) in the proof of 6.3.3 holds when a < a < b c. Since gðxÞ ! 1, we may also assume that C06 12/09/2010 14:54:7 Page 185 6.3 L’HOSPITAL’S RULES 185 gðcÞ > 0. Taking b ¼ c in (3), we have ð6Þ Le< f ð cÞ f ð a Þ <Lþe gð c Þ gð a Þ for a 2 ða; cÞ: Since gðcÞ=gðaÞ ! 0 as a ! aþ, we may assume that 0 < gðcÞ=gðaÞ < 1 for all a 2 ða; cÞ, whence it follows that gð a Þ gð c Þ gð c Þ ¼1 >0 gð a Þ gð a Þ for a 2 ða; cÞ: If we multiply (6) by ðgðaÞ gðcÞÞ=gðaÞ > 0, we have gð c Þ f ðaÞ f ðcÞ gð c Þ < ðL þ eÞ 1 ð7Þ ðL eÞ 1 < : gðaÞ gðaÞ gðaÞ gðaÞ Now, since gðcÞ=gðaÞ ! 0 and f ðcÞ=gðaÞ ! 0 as a ! aþ, then for any d with 0 < d < 1 there exists d 2 ða; cÞ such that 0 < gðcÞ=gðaÞ < d and j f ðcÞj=gðaÞ < d for all a 2 ða; d Þ, whence (7) gives ð8Þ ðL eÞð1 dÞ d < f ða Þ < ðL þ eÞ þ d: gðaÞ If we take d :¼ minf1; e; e=ðjLj þ 1Þg, it is an exercise to show that L 2e f ða Þ L þ 2e: gð a Þ Since e > 0 is arbitrary, this yields the assertion. The cases L ¼ 0 and L < 0 are handled similarly. Case (b): If L ¼ þ1, let M > 1 be given and c 2 ða; bÞ be such that f 0 ðuÞ=g0 ðuÞ > M for all u 2 ða; cÞ. Then it follows as before that f ð bÞ f ð a Þ >M gð b Þ gð a Þ ð9Þ for a < a < b c: Since gðxÞ ! 1 as x ! aþ, we may suppose that c also satisfies gðcÞ > 0, that j f ðcÞj=gðaÞ < 12, and that 0 < gðcÞ=gðaÞ < 12 for all a 2 ða; cÞ. If we take b ¼ c in (9) and multiply by 1 gðcÞ=gðaÞ > 12 , we get f ðaÞ f ðcÞ gð c Þ >M 1 > 12 M; gð a Þ gð a Þ so that f ða Þ 1 f ð cÞ 1 > 2Mþ > ðM1Þ gðaÞ gðaÞ 2 for a 2 ða; cÞ: Since M > 1 is arbitrary, it follows that lim f ðaÞ=gðaÞ ¼ 1. a!aþ If L ¼ 1, the argument is similar. Q.E.D. ln x . x Here f ðxÞ :¼ ln x and gðxÞ :¼ x on the interval (0, 1). If we apply the left-hand ln x 1=x version of 6.3.5, we obtain lim ¼ lim ¼ 0. x!1 x x!1 1 x 2 (b) We consider lim e x . 6.3.6 Examples (a) We consider lim x!1 x!1 C06 12/09/2010 14:54:7 186 Page 186 CHAPTER 6 DIFFERENTIATION Here we take f ðxÞ :¼ x2 and gðxÞ :¼ ex on R. We obtain x2 2x 2 ¼ lim x ¼ lim x ¼ 0: x x!1 e x!1 e x!1 e ln sin x . (c) We consider lim x!0þ ln x Here we take f ðxÞ :¼ ln sin x and gðxÞ :¼ ln x on ð0; pÞ. If we apply 6.3.5, we obtain h x i ln sin x cos x=sin x lim ¼ lim ¼ lim ½cos x: x!0þ ln x x!0þ x!0þ sin x 1=x lim Since lim ½x=sin x ¼ 1 and lim cos x ¼ 1, we conclude that the limit under considerax!0þ x!0þ tion equals 1. x sin x . This has indeterminate form 1=1. An application of þ sin x L’Hospital’s Rule gives us 1 cos x ; lim x!1 1 þ cos x which is useless because this limit does not exist. (Why not?) However, if we rewrite the original limit, we get directly that (d) Consider lim x!1 x sin x x ¼ 1 0 ¼ 1: lim x!1 sin x 1 þ 0 1þ x 1 & Other Indeterminate Forms Indeterminate forms such as 1 1; 0 1; 11 ; 00 ; 10 can be reduced to the previously considered cases by algebraic manipulations and the use of the logarithmic and exponential functions. Instead of formulating these variations as theorems, we illustrate the pertinent techniques by means of examples. 6.3.7 Examples (a) Let I :¼ ð0; p=2Þ and consider 1 1 lim ; x!0þ x sin x which has the indeterminate form 1 1. We have 1 1 sin x x cos x 1 ¼ lim ¼ lim lim x!0þ x x!0þ x sin x x!0þ sin x þ x cos x sin x sin x 0 ¼ ¼ 0: ¼ lim x!0þ 2 cos x x sin x 2 (b) Let I :¼ ð0; 1Þ and consider lim x ln x, which has the indeterminate form x!0þ 0 ð1Þ. We have ln x 1=x ¼ lim ¼ lim ðxÞ ¼ 0: x!0þ 1=x x!0þ 1=x2 x!0þ lim x ln x ¼ lim x!0þ (c) Let I :¼ ð0; 1Þ and consider lim xx, which has the indeterminate form 00. x!0þ We recall from calculus (see also Section 8.3) that xx ¼ ex ln x . It follows from part (b) and the continuity of the function y 7! ey at y ¼ 0 that lim xx ¼ e0 ¼ 1. x!0þ C06 12/09/2010 14:54:8 Page 187 6.3 L’HOSPITAL’S RULES 187 (d) Let I :¼ ð1; 1Þ and consider lim ð1 þ 1=xÞx, which has the indeterminate form 11 . x!1 We note that ð1 þ 1=xÞx ¼ ex lnð1þ1=xÞ : ð10Þ Moreover, we have lnð1 þ 1=xÞ 1=x ð1 þ 1=xÞ1 ðx2 Þ 1 ¼ 1: ¼ lim ¼ lim x!1 x!1 1 þ 1=x x2 lim x lnð1 þ 1=xÞ ¼ lim x!1 x!1 Since y 7! ey is continuous at y ¼ 1, we infer that lim ð1 þ 1=xÞx ¼ e. x!1 (e) Let I :¼ ð0; 1Þ and consider lim ð1 þ 1=xÞx, which has the indeterminate form 10 . x!0þ In view of formula (10), we consider lnð1 þ 1=xÞ 1 ¼ lim ¼ 0: lim x lnð1 þ 1=xÞ ¼ lim x!0þ x!0þ x!0þ 1 þ 1=x 1=x Therefore we have lim ð1 þ 1=xÞx ¼ e0 ¼ 1. & x!0þ Exercises for Section 6.3 1. Suppose that f and g are continuous on [a, b], differentiable on (a, b), that c 2 ½a; b and that gðxÞ 6¼ 0 for x 2 ½a; b; x 6¼ c. Let A :¼ lim f and B :¼ lim g. If B ¼ 0, and if lim f ðxÞ=gðxÞ x!c x!c exists in R, show that we must have A ¼ 0. [Hint: f ðxÞ ¼ f f ðxÞ=gðxÞggðxÞ.] x!c 2. In addition to the suppositions of the preceding exercise, let gðxÞ > 0 for x 2 ½a; b; x 6¼ c. If A > 0 and B ¼ 0, prove that we must have lim f ðxÞ=gðxÞ ¼ 1. If A < 0 and B ¼ 0, prove that x!c we must have lim f ðxÞ=gðxÞ ¼ 1. x!c 3. Let f ðxÞ :¼ x2 sin ð1=xÞ for 0 < x 1 and f ð0Þ :¼ 0, and let gðxÞ :¼ x2 for x 2 ½0; 1. Then both f and g are differentiable on [0,1] and gðxÞ > 0 for x 6¼ 0. Show that lim f ðxÞ ¼ 0 ¼ x!0 lim gðxÞ and that lim f ðxÞ=gðxÞ does not exist. x!0 x!0 4. Let f ðxÞ :¼ x2 for x rational, let f ðxÞ :¼ 0 for x irrational, and let g(x) :¼ sin x for x 2 R. Use Theorem 6.3.1 to show that lim f ðxÞ=gðxÞ ¼ 0. Explain why Theorem 6.3.3 cannot be used. x!0 5. Let f ðxÞ :¼ x2 sin(l=x) for x 6¼ 0, let f ð0Þ :¼ 0, and let gðxÞ :¼ sin x for x 2 R. Show that lim f ðxÞ=gðxÞ ¼ 0 but that lim f 0 ðxÞ=g0 ðxÞ does not exist. x!0 x!0 6. Evaluate the following limits. ex þex 2 x2 sin 2 x (b) lim x!0 1 cosx x!0 x4 7. Evaluate the following limits, where the domain of the quotient is as indicated. (a) lim lnðx þ 1Þ ð0; p=2Þ, sin x ln cos x (c) lim ð0; p=2Þ, x!0þ x 8. Evaluate the following limits: Arctan x (a) lim ð1; 1Þ, x!0 x (a) (c) lim x!0þ lim x3 ln x x!0þ ð0; 1Þ, (b) (d) (b) (d) tan x ð0; p=2Þ, x tan x x ð0; p=2Þ. lim x!0þ x3 lim x!0þ lim 1 x!0 xðln xÞ2 x3 x!1 e3 lim ð0; 1Þ, ð0; 1Þ C06 12/09/2010 14:54:9 188 Page 188 CHAPTER 6 DIFFERENTIATION 9. Evaluate the following limits: ln x (a) lim 2 ð0; 1Þ, x!1 x (c) (b) lim x ln sin x ð0; pÞ, (d) x!0 ln x lim pffiffiffi ð0; 1Þ, x x þ ln x ð0; 1Þ. lim x!1 x ln x x!1 10. Evaluate the following limits: (a) (c) lim x2x x!0þ ð0; 1Þ, lim ð1 þ 3=xÞx x!1 (b) ð0; 1Þ, (d) lim ð1 þ 3=xÞx ð0; 1Þ, 1 1 ð0; 1Þ. lim x!0þ x Arctan x x!0 11. Evaluate the following limits: (a) (c) lim x1=x x!1 lim xsin x x!0þ ð0; 1Þ, (b) ð0; 1Þ, (d) lim ðsin xÞx x!0þ ð0; pÞ, lim ðsec x tan xÞ ð0; p=2Þ. x!p=2 12. Let f be differentiable on (0, 1) and suppose that lim ð f ðxÞ þ f 0 ðxÞÞ ¼ L. Show that x!1 lim f ðxÞ ¼ L and lim f 0 ðxÞ ¼ 0. [Hint: f ðxÞ ¼ ex f ðxÞ=ex .] x!1 x!1 tan x as x ! ðp=2Þ. Then evaluate directly 13. Try to use L’Hospital’s Rule to find the limit of sec x by changing to sines and cosines. 14. Show that if c > 0, then lim x!c xc cx 1 ln c . ¼ x c 1 þ ln c x c Section 6.4 Taylor’s Theorem A very useful technique in the analysis of real functions is the approximation of functions by polynomials. In this section we will prove a fundamental theorem in this area that goes back to Brook Taylor (1685–1731), although the remainder term was not provided until much later by Joseph-Louis Lagrange (1736–1813). Taylor’s Theorem is a powerful result that has many applications. We will illustrate the versatility of Taylor’s Theorem by briefly discussing some of its applications to numerical estimation, inequalities, extreme values of a function, and convex functions. Taylor’s Theorem can be regarded as an extension of the Mean Value Theorem to ‘‘higher order’’ derivatives. Whereas the Mean Value Theorem relates the values of a function and its first derivative, Taylor’s Theorem provides a relation between the values of a function and its higher order derivatives. Derivatives of order greater than one are obtained by a natural extension of the differentiation process. If the derivative f 0 ðxÞ of a function f exists at every point x in an interval I containing a point c, then we can consider the existence of the derivative of the function f 0 at the point c. In case f 0 has a derivative at the point c, we refer to the resulting number as the second derivative of f at c, and we denote this number by f 00 ðcÞ or by f (2)(c). In similar fashion we define the third derivative f 000 ðcÞ ¼ f ð3Þ ðcÞ; . . ., and the nth derivative f (n)(c), whenever these derivatives exist. It is noted that the existence of the nth derivative at c presumes the existence of the (n l)st derivative in an interval containing c, but we do allow the possibility that c might be an endpoint of such an interval. If a function f has an nth derivative at a point x0, it is not difficult to construct an ðkÞ nth degree polynomial Pn such that Pn ðx0 Þ ¼ f ðx0 Þ and PðkÞ ðx0 Þ for n ðx0 Þ ¼ f C06 12/09/2010 14:54:9 Page 189 6.4 TAYLOR’S THEOREM k ¼ 1; 2; . . . ; n. In fact, the polynomial Pn ðxÞ :¼ f ðx0 Þ þ f 0 ðx0 Þðx x0 Þ þ ð1Þ þ þ 189 f 00 ðx0 Þ ðx x0 Þ2 2! f ðnÞ ðx0 Þ ðx x0 Þn n! has the property that it and its derivatives up to order n agree with the function f and its derivatives up to order n, at the specified point x0. This polynomial Pn is called the nth Taylor polynomial for f at x0. It is natural to expect this polynomial to provide a reasonable approximation to f for points near x0, but to gauge the quality of the approximation, it is necessary to have information concerning the remainder Rn :¼ f Pn . The following fundamental result provides such information. 6.4.1 Taylor’s Theorem Let n 2 N, let I :¼ ½a; b, and let f : I ! R be such that f and its derivatives f 0 ; f 00 ; . . .; f ðnÞ are continuous on I and that f ðnþ1Þ exists on (a, b). If x0 2 I, then for any x in I there exists a point c between x and x0 such that ð2Þ f 00 ðx0 Þ ðx x0 Þ2 2! f ðnÞ ðx0 Þ f ðnþ1Þ ðcÞ ðx x0 Þn þ ðx x0 Þnþ1 : þ þ n! ðn þ 1Þ! f ðxÞ ¼ f ðx0 Þ þ f 0 ðx0 Þðx x0 Þ þ Proof. Let x0 and x be given and let J denote the closed interval with endpoints x0 and x. We define the function F on J by FðtÞ :¼ f ðxÞ f ðtÞ ðx tÞ f 0 ðtÞ ðx tÞn ðnÞ f ðtÞ n! for t 2 J. Then an easy calculation shows that we have F 0 ðtÞ ¼ If we define G on J by ðx tÞn ðnþ1Þ f ðtÞ: n! xt GðtÞ :¼ FðtÞ x x0 nþ1 Fðx0 Þ for t 2 J, then Gðx0 Þ ¼ GðxÞ ¼ 0. An application of Rolle’s Theorem 6.2.3 yields a point c between x and x0 such that 0 ¼ G0 ðcÞ ¼ F 0 ðcÞ þ ðn þ 1Þ ðx cÞn ðx x0 Þnþ1 Fðx0 Þ: Hence, we obtain 1 ðx x0 Þnþ1 0 F ðcÞ n þ 1 ðx cÞn nþ1 1 ðx x0 Þ ðx cÞn ðnþ1Þ f ðnþ1Þ ðcÞ ðx x0 Þnþ1 ; f ðcÞ ¼ ¼ n þ 1 ðx cÞn ðn þ 1Þ! n! Fðx0 Þ ¼ which implies the stated result. Q.E.D. We shall use the notation Pn for the nth Taylor polynomial (1) of f, and Rn for the remainder. Thus we may write the conclusion of Taylor’s Theorem as f ðxÞ ¼ Pn ðxÞ þ C06 12/09/2010 14:54:10 190 Page 190 CHAPTER 6 DIFFERENTIATION Rn ðxÞ where Rn is given by Rn ðxÞ :¼ ð3Þ f ðnþ1Þ ðcÞ ðx x0 Þnþ1 ðn þ 1Þ! for some point c between x and x0. This formula for Rn is referred to as the Lagrange form (or the derivative form) of the remainder. Many other expressions for Rn are known; one is in terms of integration and will be given later. (See Theorem 7.3.18.) Applications of Taylor’s Theorem The remainder term Rn in Taylor’s Theorem can be used to estimate the error in approximating a function by its Taylor polynomial Pn. If the number n is prescribed, then the question of the accuracy of the approximation arises. On the other hand, if a certain accuracy is specified, then the question of finding a suitable value of n is germane. The following examples illustrate how one responds to these questions. pffiffiffiffiffiffiffiffiffiffiffiffiffi 6.4.2 Examples (a) Use Taylor’s Theorem with n ¼ 2 to approximate 3 1 þ x; x > 1. 1=3 We take the function f ðxÞ :¼ ð1 þ xÞ , the point x0 ¼ 0, and n ¼ 2. Since f 0 ðxÞ ¼ 2=3 1 and f 00 ðxÞ ¼ 13 ð 23Þð1 þ xÞ5=3 , we have f 0 ð0Þ ¼ 13 and f 00 ð0Þ ¼ 2=9. Thus 3 ð1 þ xÞ we obtain f ðxÞ ¼ P2 ðxÞ þ R2 ðxÞ ¼ 1 þ 13 x 19 x2 þ R2 ðxÞ; 5 ð1 þ cÞ8=3 x3 for some point c between 0 and x. where R2 ðxÞ ¼ 3!1 f 000 ðcÞx3 ¼ 81 pffiffiffiffiffiffiffiffi For example, if we let x ¼ 0.3, we get the approximation P2(0.3) ¼ 1.09 for 3 1:3: Moreover, since c > 0 in this case, then ð1 þ cÞ8=3 < 1 and so the error is at most 5 3 3 1 < 0:17 102 : R2 ð0:3Þ ¼ 81 10 600 pffiffiffiffiffiffiffi Hence, we have j 3 1:31:09j < 0:5 102 , so that two decimal place accuracy is assured. (b) Approximate the number e with error less than 105 . We shall consider the function gðxÞ :¼ ex and take x0 ¼ 0 and x ¼ 1 in Taylor’s Theorem. We need to determine n so that jRn ð1Þj < 105 . To do so, we shall use the fact that g0 ðxÞ ¼ ex and the initial bound of ex 3 for 0 x 1. Since g0 ðxÞ ¼ ex , it follows that gðkÞ ðxÞ ¼ ex for all k 2 N, and therefore gðkÞ ð0Þ ¼ 1 for all k 2 N. Consequently the nth Taylor polynomial is given by x2 xn þ þ 2! n! c and the remainder for x ¼ 1 is given by Rn ð1Þ ¼ e =ðn þ 1Þ! for some c satisfying 0 < c < 1. Since ec < 3, we seek a value of n such that 3=ðn þ 1Þ! < 105 . A calculation reveals that 9! ¼ 362; 880 > 3 105 so that the value n ¼ 8 will provide the desired accuracy; moreover, since 8! ¼ 40; 320, no smaller value of n will be certain to suffice. Thus, we obtain Pn ðxÞ :¼ 1 þ x þ e P8 ð1Þ ¼ 1 þ 1 þ 1 1 þ þ ¼ 2:718 28 2! 8! with error less than 105 . Taylor’s Theorem can also be used to derive inequalities. & C06 12/09/2010 14:54:10 Page 191 6.4 TAYLOR’S THEOREM 191 6.4.3 Examples (a) 1 12 x2 cos x for all x 2 R. Use f ðxÞ :¼ cos x and x0 ¼ 0 in Taylor’s Theorem, to obtain 1 cos x ¼ 1 x2 þ R2 ðxÞ; 2 where for some c between 0 and x we have f 000 ðcÞ 3 sin c 3 x ¼ x : R2 ðxÞ ¼ 3! 6 If 0 x p, then 0 c < p; since c and x3 are both positive, we have R2 ðxÞ 0. Also, if p x 0, then p c 0; since sin c and x3 are both negative, we again have R2 ðxÞ 0. Therefore, we see that 1 12 x2 cos x for jxj p. If jxj p, then we have 1 12 x2 < 3 cos x and the inequality is trivially valid. Hence, the inequality holds for all x 2 R. (b) For any k 2 N, and for all x > 0, we have 1 1 1 1 x2kþ1 : x x2 þ x2k < lnð1 þ xÞ < x x2 þ þ 2 2k 2 2k þ 1 Using the fact that the derivative of ln(l þ x) is 1=(1 þ x) for x > 0, we see that the nth Taylor polynomial for ln(l þ x) with x0 ¼ 0 is 1 1 Pn ðxÞ ¼ x x2 þ þ ð1Þn1 xn 2 n and the remainder is given by ð1Þn cnþ1 nþ1 x nþ1 for some c satisfying 0 < c < x. Thus for any x > 0, if n ¼ 2k is even, then we have R2k ðxÞ > 0; and if n ¼ 2k þ 1 is odd, then we have R2kþ1 ðxÞ < 0. The stated inequality then follows immediately. (c) ep > pe Taylor’s Theorem gives us the inequality ex > 1 þ x for x > 0, which the reader should verify. Then, since p > e, we have x ¼ p=e 1 > 0, so that Rn ðxÞ ¼ This implies e p=e eðp=e1Þ > 1 þ ðp=e 1Þ ¼ p=e: > ðp=eÞe ¼ p, and thus we obtain the inequality ep > pe. & Relative Extrema It was established in Theorem 6.2.1 that if a function f : I ! R is differentiable at a point c interior to the interval I, then a necessary condition for f to have a relative extremum at c is that f 0 ðcÞ ¼ 0. One way to determine whether f has a relative maximum or relative minimum [or neither] at c is to use the First Derivative Test 6.2.8. Higher order derivatives, if they exist, can also be used in this determination, as we now show. 6.4.4 Theorem Let I be an interval, let x0 be an interior point of I, and let n 2. Suppose that the derivatives f 0 ; f 00 . . . ; f ðnÞ exist and are continuous in a neighborhood of x0 and that f 0 ðx0 Þ ¼ ¼ f ðn1Þ ðx0 Þ ¼ 0, but f ðnÞ ðx0 Þ 6¼ 0. (i) (ii) (iii) If n is even and f ðnÞ ðx0 Þ > 0, then f has a relative minimum at x0. If n is even and f ðnÞ ðx0 Þ < 0, then f has a relative maximum at x0. If n is odd, then f has neither a relative minimum nor relative maximum at x0. C06 12/09/2010 14:54:11 192 Page 192 CHAPTER 6 DIFFERENTIATION Proof. Applying Taylor’s Theorem at x0, we find that for x 2 I we have f ðnÞ ðcÞ ðx x0 Þn ; n! where c is some point between x0 and x. Since f (n) is continuous, if f ðnÞ ðx0 Þ 6¼ 0, then there exists an interval U containing x0 such that f ðnÞ ðxÞ will have the same sign as f ðnÞ ðx0 Þ for x 2 U. If x 2 U, then the point c also belongs to U and consequently f ðnÞ ðcÞ and f ðnÞ ðx0 Þ will have the same sign. f ðxÞ ¼ Pn1 ðxÞ þ Rn1 ðxÞ ¼ f ðx0 Þ þ (i) If n is even and f ðnÞ ðx0 Þ > 0, then for x 2 U we have f ðnÞ ðcÞ > 0 and ðx x0 Þn 0 so that Rn1 ðxÞ 0. Hence, f ðxÞ f ðx0 Þ for x 2 U, and therefore f has a relative minimum at x0. (ii) If n is even and f ðnÞ ðx0 Þ < 0, then it follows that Rn1 ðxÞ 0 for x 2 U, so that f ðxÞ f ðx0 Þ for x 2 U. Therefore, f has a relative maximum at x0. (iii) If n is odd, then ðx x0 Þn is positive if x > x0 and negative if x < x0 . Consequently, if x 2 U, then Rn1 ðxÞ will have opposite signs to the left and to the right of x0. Therefore, f has neither a relative minimum nor a relative maximum at x0 . Q.E.D. Convex Functions The notion of convexity plays an important role in a number of areas, particularly in the modern theory of optimization. We shall briefly look at convex functions of one real variable and their relation to differentiation. The basic results, when appropriately modified, can be extended to higher dimensional spaces. 6.4.5 Definition Let I R be an interval. A function f : I ! R is said to be convex on I if for any t satisfying 0 t 1 and any points x1 ; x2 in I, we have f ðð1 tÞx1 þ tx2 Þ ð1 tÞ f ðx1 Þ þ tf ðx2 Þ: Note that if x1 < x2 , then as t ranges from 0 to 1, the point ð1 tÞx1 þ tx2 traverses the interval from x1 to x2 . Thus if f is convex on I and if x1 ; x2 2 I, then the chord joining any two points ðx1 ; f ðx1 ÞÞ and ðx2 ; f ðx2 ÞÞ on the graph of f lies above the graph of f. (See Figure 6.4.1.) Figure 6.4.1 A convex function A convex function need not be differentiable at every point, as the example f ðxÞ :¼ jxj; x 2 R, reveals. However, it can be shown that if I is an open interval and if f : I ! R is convex on I, then the left and right derivatives of f exist at every point of I. As a C06 12/09/2010 14:54:12 Page 193 6.4 TAYLOR’S THEOREM 193 consequence, it follows that a convex function on an open interval is necessarily continuous. We will not verify the preceding assertions, nor will we develop many other interesting properties of convex functions. Rather, we will restrict ourselves to establishing the connection between a convex function f and its second derivative f 00 , assuming that f 00 exists. 6.4.6 Theorem Let I be an open interval and let f : I ! R have a second derivative on I. Then f is a convex function on I if and only if f 00 ðxÞ 0 for all x 2 I. Proof. ()) We will make use of the fact that the second derivative is given by the limit f 00 ðaÞ ¼ lim ð4Þ h!0 f ða þ hÞ 2f ðaÞ þ f ða hÞ h2 for each a 2 I. (See Exercise 16.) Given a 2 I, let h be such that a þ h and a h belong to I. Then a ¼ 12 ðða þ hÞ þ ða hÞÞ, and since f is convex on I, we have f ðaÞ ¼ f 12ða þ hÞ þ 12ða hÞ 12 f ða þ hÞ þ 12 f ða hÞ: Therefore, we have f ða þ hÞ 2f ðaÞ þ f ða hÞ 0. Since h2 > 0 for all h 6¼ 0, we see that the limit in (4) must be nonnegative. Hence, we obtain f 00 ðaÞ 0 for any a 2 I. ð(Þ We will use Taylor’s Theorem. Let x1 ; x2 be any two points of I, let 0 < t < 1, and let x0 :¼ ð1 tÞx1 þ tx2 . Applying Taylor’s Theorem to f at x0 we obtain a point cl between x0 and x1 such that f ðx1 Þ ¼ f ðx0 Þ þ f 0 ðx0 Þðx1 x0 Þ þ 12 f 00 ðc1 Þðx1 x0 Þ2 ; and a point c2 between x0 and x2 such that f ðx2 Þ ¼ f ðx0 Þ þ f 0 ðx0 Þðx2 x0 Þ þ 12 f 00 ðc2 Þðx2 x0 Þ2 : If f 00 is nonnegative on I, then the term R :¼ 12ð1 tÞf 00 ðc1 Þðx1 x0 Þ2 þ 12 tf 00 ðc2 Þðx2 x0 Þ2 is also nonnegative. Thus we obtain ð1 tÞ f ðx1 Þ þ tf ðx2 Þ ¼ f ðx0 Þ þ f 0 ðx0 Þðð1 tÞx1 þ tx2 x0 Þ þ 12ð1 tÞf 00 ðc1 Þðx1 x0 Þ2 þ 12 tf 00 ðc2 Þðx2 x0 Þ2 ¼ f ðx0 Þ þ R f ðx0 Þ ¼ f ðð1 tÞx1 þ tx2 Þ: Hence, f is a convex function on I. Q.E.D. Newton’s Method It is often desirable to estimate a solution of an equation with a high degree of accuracy. The Bisection Method, used in the proof of the Location of Roots Theorem 5.3.5, provides one estimation procedure, but it has the disadvantage of converging to a solution rather slowly. A method that often results in much more rapid convergence is based on the geometric idea of successively approximating a curve by tangent lines. The method is named after its discoverer, Isaac Newton. C06 12/09/2010 14:54:12 194 Page 194 CHAPTER 6 DIFFERENTIATION Let f be a differentiable function that has a zero at r and let x1 be an initial estimate of r. The line tangent to the graph at ðx1 ; f ðx1 ÞÞ has the equation y ¼ f ðx1 Þ þ f 0 ðx1 Þðx x1 Þ, and crosses the x-axis at the point x2 :¼ x1 f ðx1 Þ : f 0 ðx1 Þ (See Figure 6.4.2.) If we replace x1 by the second estimate x2 , then we obtain a point x3 , and so on. At the nth iteration we get the point xnþ1 from the point xn by the formula xnþ1 :¼ xn f ðxn Þ : f 0 ðxn Þ Under suitable hypotheses, the sequence ðxn Þ will converge rapidly to a root of the equation f ðxÞ ¼ 0, as we now show. The key tool in establishing the rapid rate of convergence is Taylor’s Theorem. Figure 6.4.2 Newton’s Method 6.4.7 Newton’s Method Let I :¼ ½a; b and let f : I ! R be twice differentiable on I. Suppose that f ðaÞ f ðbÞ < 0 and that there are constants m, M such that j f 0 ðxÞj m > 0 and j f 00 ðxÞj M for x 2 I and let K :¼ M=2m. Then there exists a subinterval I containing a zero r of f such that for any x1 2 I the sequence ðxn Þ defined by ð5Þ xnþ1 :¼ xn f ðxn Þ f 0 ðxn Þ f or all n 2 N; belongs to I and ðxn Þ converges to r. Moreover ð6Þ jxnþ1 rj K jxn rj2 f or all n 2 N: Proof. Since f ðaÞ f ðbÞ < 0, the numbers f (a) and f (b) have opposite signs; hence by Theorem 5.3.5 there exists r 2 I such that f ðrÞ ¼ 0. Since f 0 is never zero on I, it follows from Rolle’s Theorem 6.2.3 that f does not vanish at any other point of I. We now let x0 2 I be arbitrary; by Taylor’s Theorem there exists a point c0 between x0 and r such that 0 ¼ f ðrÞ ¼ f ðx0 Þ þ f 0 ðx0 Þðr x0 Þ þ 12 f 00 ðc0 Þðr x0 Þ2 ; C06 12/09/2010 14:54:13 Page 195 6.4 TAYLOR’S THEOREM 195 from which it follows that f ðx0 Þ ¼ f 0 ðx0 Þðr x0 Þ þ 12 f 00 ðc0 Þðr x0 Þ2 : If x00 is the number defined from x0 by ‘‘the Newton procedure’’: x00 :¼ x0 f ðx0 Þ ; f 0 ðx0 Þ then an elementary calculation shows that x00 ¼ x0 þ ðr x0 Þ þ 1 f 00 ðc0 Þ ðr x0 Þ2 ; 2 f 0 ðx0 Þ whence it follows that x00 r ¼ 1 f 00 ðc0 Þ 0 ðx rÞ2 : 2 f 0 ðx0 Þ Since c0 2 I, the assumed bounds on f 0 and f 00 hold and, setting K :¼ M=2m, we obtain the inequality ð7Þ jx00 rj K jx0 rj2 : We now choose d > 0 so small that d < 1=K and that the interval I :¼ ½r d; r þ d is contained in I. If xn 2 I , then jxn rj d and it follows from (7) that jxnþ1 rj Kjxn rj2 Kd2 < d; hence xn 2 I implies that xnþ1 2 I . Therefore if x1 2 I , we infer that xn 2 I for all n 2 N. Also if x1 2 I , then an elementary induction argument using (7) shows that jxnþ1 rj < ðKdÞn jx1 rj for n 2 N. But since Kd < 1 this proves that limðxn Þ ¼ r. Q.E.D. pffiffiffi 6.4.8 Example We will illustrate Newton’s Method by using it to approximate 2. If we let f ðxÞ :¼ x2 2 for x 2 R, then we seek the positive root of the equation f ðxÞ ¼ 0. Since f 0 ðxÞ ¼ 2x, the iteration formula is xnþ1 ¼ xn f ðxn Þ f 0 ðxn Þ x2n 2 1 2 xn þ ¼ ¼ xn : 2xn 2 xn If we take x1 :¼ 1 as our initial estimate, we obtain the successive values x2 ¼ 3=2 ¼ 1:5, x3 ¼ 17=12 ¼ 1:416 666 . . ., x4 ¼ 577=408 ¼ 1:414 215 . . ., and x5 ¼ 665 857=470 832 & ¼ 1:414 213 562 374 . . ., which is correct to eleven places. Remarks (a) If we let en :¼ xn r be the error in approximating r, then inequality (6) can be written in the form jKenþ1 j jKen j2 . Consequently, if jKen j < 10m then jKenþ1 j < 102m so that the number of significant digits in Ken has been doubled. Because of this doubling, the sequence generated by Newton’s Method is said to converge ‘‘quadratically.’’ (b) In practice, when Newton’s Method is programmed for a computer, one often makes an initial guess x1 and lets the computer run. If x1 is poorly chosen, or if the root is too near the endpoint of I, the procedure may not converge to a zero of f. Two possible difficulties C06 12/09/2010 14:54:14 196 Page 196 CHAPTER 6 DIFFERENTIATION are illustrated in Figures 6.4.3 and 6.4.4. One familiar strategy is to use the Bisection Method to arrive at a fairly close estimate of the root and then to switch to Newton’s Method for the coup de gr^ace. Figure 6.4.3 Figure 6.4.4 xn oscillates between x1 and x2 xn ! 1 Exercises for Section 6.4 1. Let f ðxÞ :¼ cos ax for x 2 R where a 6¼ 0. Find f ðnÞ ðxÞ for n 2 N, x 2 R. 2. Let gðxÞ :¼ jx3 j for x 2 R. Find g0 ðxÞ and g00 ðxÞ for x 2 R, and g000 ðxÞ for x 6¼ 0. Show that g000 ð0Þ does not exist. 3. Use Induction to prove Leibniz’s rule for the nth derivative of a product: n X n ðnkÞ f ðxÞgðkÞ ðxÞ: ð f gÞðnÞ ðxÞ ¼ k k¼0 pffiffiffiffiffiffiffiffiffiffiffi 1 þ x 1 þ 12 x. pffiffiffiffiffiffiffi pffiffiffi 5. Use the preceding exercise to approximate 1:2 and 2. What is the best accuracy you can be sure of, using this inequality? pffiffiffiffiffiffiffi pffiffiffi 6. Use Taylor’s Theorem with n ¼ 2 to obtain more accurate approximations for 1:2 and 2. 4. Show that if x > 0, then 1 þ 12 x 18 x2 1=3 1 1 2 3 7. If x >p0ffiffiffiffiffiffi show that p ffiffiffijð1 þ xÞ ð1 þ 3 x 9 x Þj ð5=81Þx . Use this inequality to approxiffi 3 3 mate 1:2 and 2. 8. If f ðxÞ :¼ ex , show that the remainder term in Taylor’s Theorem converges to zero as n ! 1, for each fixed x0 and x. [Hint: See Theorem 3.2.11.] 9. If gðxÞ :¼ sin x, show that the remainder term in Taylor’s Theorem converges to zero as n ! 1 for each fixed x0 and x. 10. Let hðxÞ :¼ e1=x for x 6¼ 0 and hð0Þ :¼ 0. Show that hðnÞ ð0Þ ¼ 0 for all n 2 N. Conclude that the remainder term in Taylor’s Theorem for x0 ¼ 0 does not converge to zero as n ! 1 for x 6¼ 0. [Hint: By L’Hospital’s Rule, lim hðxÞ=xk ¼ 0 for any k 2 N. Use Exercise 3 to calculate x!0 hðnÞ ðxÞ for x 6¼ 0.] 2 11. If x 2 ½0; 1 and n 2 N, show that 2 3 n nþ1 lnð1 þ xÞ x x þ x þ þ ð1Þn1 x < x : 2 3 n nþ1 Use this to approximate ln 1.5 with an error less than 0.01. Less than 0.001. C06 12/09/2010 14:54:14 Page 197 6.4 TAYLOR’S THEOREM 197 12. We wish to approximate sine by a polynomial on ½1; 1 so that the error is less than 0.001. Show that we have 3 5 sin x x x þ x < 1 for jxj 1: 6 120 5040 13. Calculate e correct to seven decimal places. 14. Determine whether or not x ¼ 0 is a point of relative extremum of the following functions: (a) f ðxÞ :¼ x3 þ 2; (b) gðxÞ :¼ sin x x; (c) hðxÞ :¼ sin x þ 16 x3 ; (d) kðxÞ :¼ cos x 1 þ 12 x2 : 15. Let f be continuous on [a, b] and assume the second derivative f 00 exists on (a, b). Suppose that the graph of f and the line segment joining the points ða; f ðaÞÞ and ðb; f ðbÞÞ intersect at a point ðx0 ; f ðx0 ÞÞ where a < x0 < b. Show that there exists a point c 2 ða; bÞ such that f 00 ðcÞ ¼ 0. 16. Let I R be an open interval, let f : I ! R be differentiable on I, and suppose f 00 ðaÞ exists at a 2 I. Show that f 00 ðaÞ ¼ lim h!0 f ða þ hÞ 2 f ðaÞ þ f ða hÞ : h2 Give an example where this limit exists, but the function does not have a second derivative at a. 17. Suppose that I R is an open interval and that f 00 ðxÞ 0 for all x 2 I. If c 2 I, show that the part of the graph of f on I is never below the tangent line to the graph at (c, f (c)). 18. Let I R be an interval and let c 2 I. Suppose that f and g are defined on I and that the derivatives f ðnÞ ; gðnÞ exist and are continuous on I. If f ðkÞ ðcÞ ¼ 0 and gðkÞ ðcÞ ¼ 0 for k ¼ 0; 1; . . . ; n 1, but gðnÞ ðcÞ 6¼ 0, show that lim x!c f ðxÞ f ðnÞ ðcÞ ¼ : gðxÞ gðnÞ ðcÞ 19. Show that the function f ðxÞ :¼ x3 2x 5 has a zero r in the interval I :¼ ½2; 2:2. If x1 :¼ 2 and if we define the sequence (xn) using the Newton procedure, show that jxnþ1 rj ð0:7Þjxn rj2 . Show that x4 is accurate to within six decimal places. 20. Approximate the real zeros of gðxÞ :¼ x4 x 3. 21. Approximate the real zeros of hðxÞ :¼ x3 x 1. Apply Newton’s Method starting with the initial choices (a) x1 :¼ 2, (b) x1 :¼ 0, (c) x1 :¼ 2. Explain what happens. 22. The equation ln x ¼ x 2 has two solutions. Approximate them using Newton’s Method. What happens if x1 :¼ 12 is the initial point? 23. The function f ðxÞ ¼ 8x3 8x2 þ 1 has two zeros in [0,1]. Approximate them, using Newton’s Method, with the starting points (a) x1 :¼ 18, (b) x1 :¼ 14. Explain what happens. 24. Approximate the solution of the equation x ¼ cos x, accurate to within six decimals. C07 12/13/2010 10:5:29 Page 198 CHAPTER 7 THE RIEMANN INTEGRAL We have already mentioned the developments, during the 1630s, by Fermat and Descartes leading to analytic geometry and the theory of the derivative. However, the subject we know as calculus did not begin to take shape until the late 1660s when Isaac Newton created his theory of ‘‘fluxions’’ and invented the method of ‘‘inverse tangents’’ to find areas under curves. The reversal of the process for finding tangent lines to find areas was also discovered in the 1680s by Gottfried Leibniz, who was unaware of Newton’s unpublished work and who arrived at the discovery by a very different route. Leibniz introduced the terminology ‘‘calculus differentialis’’ and ‘‘calculus integralis,’’ since finding tangent lines involved differences and finding areas involved summations. Thus, they had discovered that integration, being a process of summation, was inverse to the operation of differentiation. During a century and a half of development and refinement of techniques, calculus consisted of these paired operations and their applications, primarily to physical problems. In the 1850s, Bernhard Riemann adopted a new and different viewpoint. He separated the concept of integration from its companion, differentiation, and examined the motivating summation and limit process of finding areas by itself. He broadened the scope by considering all functions on an interval for which this process of ‘‘integration’’ could be defined: the class of ‘‘integrable’’ functions. The Fundamental Theorem of Calculus became a result that held only for a restricted set of integrable functions. The viewpoint of Riemann led others to invent other integration theories, the most significant being Lebesgue’s theory of integration. But there have been some advances made in more recent times that extend even the Lebesgue theory to a considerable extent. We will give a brief introduction to these results in Chapter 10. Bernhard Riemann (Georg Friedrich) Bernhard Riemann (1826–1866), the son of a poor Lutheran minister, was born near Hanover, Germany. To please his father, he enrolled (1846) at the University of G€ ottingen as a student of theology and philosophy, but soon switched to mathematics. He interrupted his studies at G€ottingen to study at Berlin under C. G. J. Jacobi, P. G. J. Dirichlet, and F. G. Eisenstein, but returned to G€ ottingen in 1849 to complete his thesis under Gauss. His thesis dealt with what are now called ‘‘Riemann surfaces.’’ Gauss was so enthusiastic about Riemann’s work that he arranged for him to become a privatdozent at G€ ottingen in 1854. On admission as a privatdozent, Riemann was required to prove himself by delivering a probationary lecture before the entire faculty. As tradition dictated, he submitted three topics, the first two of which he was well prepared to discuss. To Riemann’s surprise, Gauss chose that he should lecture on the third topic: ‘‘On the hypotheses that underlie the foundations of geometry.’’ After its publication, this lecture had a profound effect on modern geometry. Despite the fact that Riemann contracted tuberculosis and died at the age of 39, he made major contributions in many areas: the foundations of geometry, number theory, real and complex analysis, topology, and mathematical physics. 198 C07 12/13/2010 10:5:29 Page 199 7.1 RIEMANN INTEGRAL 199 We begin by defining the concept of Riemann integrability of real-valued functions defined on a closed bounded interval of R, using the Riemann sums familiar to the reader from calculus. This method has the advantage that it extends immediately to the case of functions whose values are complex numbers, or vectors in the space R n. In Section 7.2, we will establish the Riemann integrability of several important classes of functions: step functions, continuous functions, and monotone functions. However, we will also see that there are functions that are not Riemann integrable. The Fundamental Theorem of Calculus is the principal result in Section 7.3. We will present it in a form that is slightly more general than is customary and does not require the function to be a derivative at every point of the interval. A number of important consequences of the Fundamental Theorem are also given. In Section 7.3 we also give a statement of the definitive Lebesgue Criterion for Riemann integrability. This famous result is usually not given in books at this level, since its proof (given in Appendix C) is somewhat complicated. However, its statement is well within the reach of students, who will also comprehend the power of this result. In Section 7.4, we discuss an alternative approach to the Riemann integral due to Gaston Darboux that uses the concepts of upper integral and lower integral. The two approaches appear to be quite different, but in fact they are shown to be equivalent. The final section presents several methods of approximating integrals, a subject that has become increasingly important during this era of high-speed computers. While the proofs of these results are not particularly difficult, we defer them to Appendix D. An interesting history of integration theory, including a chapter on the Riemann integral, is given in the book by Hawkins cited in the References. Section 7.1 Riemann Integral We will follow the procedure commonly used in calculus courses and define the Riemann integral as a kind of limit of the Riemann sums as the norm of the partitions tend to 0. Since we assume that the reader is familiar—at least informally—with the integral from a calculus course, we will not provide a motivation of the integral, or discuss its interpretation as the ‘‘area under the graph,’’ or its many applications to physics, engineering, economics, etc. Instead, we will focus on the purely mathematical aspects of the integral. However, we first define some basic terms that will be frequently used. Partitions and Tagged Partitions If I :¼ ½a; b is a closed bounded interval in R, then a partition of I is a finite, ordered set P :¼ ðx0 ; x1 ; . . . ; xn1 ; xn Þ of points in I such that a ¼ x0 < x1 < < xn1 < xn ¼ b: (See Figure 7.1.1.) The points of P are used to divide I ¼ [a, b] into non-overlapping subintervals I 1 :¼ ½x0 ; x1 ; I 2 :¼ ½x1 ; x2 ; . . . ; Figure 7.1.1 I n :¼ ½xn1 ; xn : A partition of [a, b] C07 12/13/2010 10:5:30 200 Page 200 CHAPTER 7 THE RIEMANN INTEGRAL Often we will denote the partition P by the notation P ¼ f½xi1 ; xi gni¼1 . We define the norm (or mesh) of P to be the number ð1Þ jjPjj :¼ maxfx1 x0 ; x2 x1 ; . . . ; xn xn1 g: Thus the norm of a partition is merely the length of the largest subinterval into which the partition divides [a, b]. Clearly, many partitions have the same norm, so the partition is not a function of the norm. If a point ti has been selected from each subinterval I i ¼ ½xi1 ; xi , for i ¼ 1; 2; . . . ; n, then the points are called tags of the subintervals I i . A set of ordered pairs P_ :¼ fð½xi1 ; xi ; ti Þgni¼1 of subintervals and corresponding tags is called a tagged partition of I; see Figure 7.1.2. (The dot over the P indicates that a tag has been chosen for each subinterval.) The tags can be chosen in a wholly arbitrary fashion; for example, we can choose the tags to be the left endpoints, or the right endpoints, or the midpoints of the subintervals, etc. Note that an endpoint of a subinterval can be used as a tag for two consecutive subintervals. Since each tag can be chosen in infinitely many ways, each partition can be tagged in infinitely many ways. The norm of a tagged partition is defined as for an ordinary partition and does not depend on the choice of tags. Figure 7.1.2 A tagged partition of [a, b] If P_ is the tagged partition given above, we define the Riemann sum of a function f : ½a; b ! R corresponding to P_ to be the number ð2Þ n X S f ; P :¼ f ðti Þðxi xi1 Þ: i¼1 We will also use this notation when P_ denotes a subset of a partition, and not the entire partition. The reader will perceive that if the function f is positive on [a, b], then the Riemann sum (2) is the sum of the areas of n rectangles whose bases are the subintervals I i ¼ ½xi1 ; xi and whose heights are f ðti Þ. (See Figure 7.1.3.) Figure 7.1.3 A Riemann sum C07 12/13/2010 10:5:30 Page 201 7.1 RIEMANN INTEGRAL 201 Definition of the Riemann Integral We now define the Riemann integral of a function f on an interval [a, b]. 7.1.1 Definition A function f : ½a; b ! R is said to be Riemann integrable on [a, b] if there exists a number L 2 R such that for every e > 0 there exists de > 0 such that if P_ is _ < de , then any tagged partition of [a, b] with jjPjj S f ; P L < e: The set of all Riemann integrable functions on [a, b] will be denoted by R½a; b. Remark It is sometimes said that the integral L is ‘‘the limit’’ of the Riemann sums _ ! 0. However, since S f ; P_ is not a function of jjPjj, _ this limit S f : P_ as the norm jjPjj is not of the type that we have studied before. First we will show that if f 2 R½a; b, then the number L is uniquely determined. It will be called the Riemann integral of f over [a, b]. Instead of L, we will usually write Z b Z b L¼ f or f ðxÞdx: a a It should be understood that any letter other than x can be used in the latter expression, so long as it does not cause any ambiguity. 7.1.2 Theorem If f 2 R½a; b, then the value of the integral is uniquely determined. Proof. Assume that L0 and L00 both satisfy the definition and let e > 0. Then there exists d0e=2 > 0 such that if P_ 1 is any tagged partition with jjP_ 1 jj < d0e=2 , then S f ; P_ 1 L0 < e=2: Also there exists d00e=2 > 0 such that if P_ 2 is any tagged partition with jjP_ 2 jj < d00e=2 , then S f ; P 2 L00 < e=2: _ < de . Since Now let de :¼ min d0e=2 ; d00e=2 > 0 and let P_ be a tagged partition with jjPjj 0 00 _ < d e=2 , then _ < de=2 and jjPjj both jjPjj S f ; P L0 < e=2 S f ; P L00 < e=2; and whence it follows from the Triangle Inequality that jL0 L00 j ¼ L0 S f ; P þ S f ; P L00 L0 S f ; P þ S f ; P L00 < e=2 þ e=2 ¼ e: Since e > 0 is arbitrary, it follows that L0 ¼ L00 . Q.E.D. The next result shows that changing a function at a finite number of points does not affect its integrability nor the value of its integral. 7.1.3 Theorem If g is Riemann integrable on [a, b] and if f ðxRÞ ¼ gðxRÞ except for a finite b b number of points in [a, b], then f is Riemann integrable and a f ¼ a g. C07 12/13/2010 10:5:31 202 Page 202 CHAPTER 7 THE RIEMANN INTEGRAL Proof. If we prove the assertion for the case of one exceptional point, then the extension to a finite number of points is done by a standard induction argument, which we leave to the reader. Rb Let c be a point in the interval and let L ¼ a g. Assume that f ðxÞ ¼ gðxÞ for all x 6¼ c. _ the terms in the two sums S f ; P_ and S g; P_ are identical For any tagged partition P, with the exception of at most two terms (in the case that c ¼ xi ¼ xi1 is an endpoint). Therefore, we have S f ; P S g; P ¼ jSð f ðxi Þ gðxi ÞÞðxi xi1 Þj 2 jgðcÞjþj f ðcÞjÞ jjPjj: Now, given e > 0, we letd1 > 0 satisfy d1 < e=ð4ðj f ðcÞj þ jgðcÞjÞ, and let d2 > 0 be such _ < d2 implies S g; P_ L < e=2. We now let d :¼ minfd1 ; d2 g. Then, if that jjPjj _ < d, we obtain jjPjj S f ; P L S f ; P S g; P þ S g; P L < e=2 þ e=2 ¼ e: Hence, the function f is integrable with integral L. Q.E.D. Some Examples If we use only the definition, in order to show that a function f is Riemann integrable we must (i) know (or guess correctly) the value L of the integral, and (ii) construct a de that will suffice for an arbitrary e > 0. The determination of L is sometimes done by calculating Riemann sums and guessing what L must be. The determination of de is likely to be difficult. In actual practice, we usually show that f 2 R½a; b by making use of some of the theorems that will be given later. 7.1.4 Examples (a) Every constant function on [a, b] is in R½a; b. Let f ðxÞ :¼ k for all x 2 ½a; b. If P_ :¼ fð½xi1 ; xi ; ti Þgni¼1 is any tagged partition of [a, b], then it is clear that n X kðxi xi1 Þ ¼ kðb aÞ: S f; P ¼ i¼1 _ < de , then Hence, for any e > 0, we can choose de :¼ 1 so that if jjPjj S f ; P kðb aÞ ¼ 0 < e: Rb Since e > 0 is arbitrary, we conclude that f 2 R½a; b and a f ¼ kðb aÞ. (b) Let g : ½0; 3 ! R be defined by gðxÞ :¼ 2 for 0 x 1, and gðxÞ :¼ 3 for 1 < x 3. A preliminary investigation, based on the graph of g (see Figure 7.1.4), suggests R3 that we might expect that 0 g ¼ 8. Figure 7.1.4 Graph of g C07 12/13/2010 10:5:31 Page 203 7.1 RIEMANN INTEGRAL 203 Let P_ be a tagged partition of [0,3] with norm < d; we will show how to determine d in order to ensure that S g; P_ 8j < e. Let P_ 1 be the subset of P_ having its tags in [0,1] where g(x) ¼ 2, and let P_ 2 be the subset of P_ with its tags in (1, 3] where g(x) ¼ 3. It is obvious that we have ð3Þ S g; P ¼ S g; P 1 þ S g; P 2 : If we let U 1 denote the union of the subintervals in P_ 1 , then it is readily shown that ð4Þ ½0; 1 d U 1 ½0; 1 þ d: For example, to prove the first inclusion, we let u 2 ½0; 1 d. Then u lies in an interval _ < d, we have xk xk1 < d. Then xk1 u I k :¼ ½xk1 ; xk of P_ 1 , and since jjPjj 1 d implies that xk xk1 þ d ð1 dÞ þ d 1. Thus the tag tk in I k satisfies tk 1 and therefore u belongs to a subinterval whose tag is in [0,1], that is, u 2 U 1 . This proves the first inclusion in (4), and the second inclusion can be shown in the same manner. Since gðtk Þ ¼ 2 for the tags of P_ 1 and since the intervals in (4) have lengths 1 d and 1 þ d, respectively, it follows that 2ð1 dÞ S g; P 1 2ð1 þ dÞ: A similar argument shows that the union of all subintervals with tags ti 2 ð1; 3 contains the interval ½1 þ d; 3 of length 2 d, and is contained in ½1 d; 3 of length 2 þ d. Therefore, 3ð2 dÞ S g; P 2 3ð2 þ dÞ: Adding these inequalities and using equation (3), we have 8 5d S g; P ¼ S g; P 1 þ S g; P 2 8 þ 5d; whence it follows that S g; P 8 5d: To have this final term < e, we are led to take de < e=5. Making such a choice (for example, if we take de :¼ e=10), we can retrace the _ < de . Since e > 0 is arbitrary, we have argument and see that S g; P_ 8 < e when jjPjj R3 proved that g 2 R½0; 3 and that 0 g ¼ 8, as predicted, (c) Let hðxÞ :¼ x for x 2 ½0; 1; we will show that h 2 R½0; 1. We will employ a ‘‘trick’’ that enables us to guess the value of the integral by considering a particular choice of the tag points. Indeed, if fI i gni¼1 is any partition of [0,1] and we choose the tag of the interval I i ¼ ½xi1 ; xi to be the midpoint qi :¼ 12 ðxi1 þ xi Þ, then the contribution of this term to the Riemann sum corresponding to the tagged partition _ :¼ fðI i ; q Þgn is Q i i¼1 1 1 hðqi Þðxi xi1 Þ ¼ ðxi þ xi1 Þðxi xi1 Þ ¼ x2i x2i1 : 2 2 If we add these terms and note that the sum telescopes, we obtain n X 1 1 1 2 xi x2i1 ¼ 12 02 ¼ : S h; Q ¼ 2 2 2 i¼1 _ < d so that Now let P_ :¼ fðI i ; ti Þgni¼1 be an arbitrary tagged partition of [0,1] with jjPjj _ xi xi1 < d for i ¼ 1; . . . ; n. Also let Q have the same partition points, but where we C07 12/13/2010 10:5:31 204 Page 204 CHAPTER 7 THE RIEMANN INTEGRAL choose the tag qi to be the midpoint of the interval I i . Since both ti and qi belong to this interval, we have jti qi j < d. Using the Triangle Inequality, we deduce n n X X _ ¼ ti ðxi xi1 Þ S h; P S h; Q qi ðxi xi1 Þ i¼1 i1 n X i¼1 n X jti qi jðxi xi1 Þ < d ðxi xi1 Þ ¼ dðxn x0 Þ ¼ d: i¼1 _ ¼ 1, we infer that if P_ is any tagged partition with jjPjj _ < d, then Since S h; Q 2 1 S h; P < d: 2 Therefore we are led to take de e. If we choose de :¼ e, we can retrace the argument to R1 R1 conclude that h 2 R½0; 1 and 0 h ¼ 0 x dx ¼ 12. (d) Let GðxÞ :¼ 1=n for x ¼ 1=n ðn 2 N Þ, and GðxÞ :¼ 0 elsewhere on ½0; 1. Given e > 0, the set E :¼ fx : GðxÞ eg is a finite set. (For example, if e ¼ 1=10, then E ¼ f1; 1=2; 1=3; . . . ; 1=10g.) If n is the number of points in E, we allow for the possibility that a tag may be counted twice if it is an endpoint and let d :¼ e=2n. For a given _ < d, we let P_ 0 be the subset of P_ with all tags outside of E tagged partition P_ such that jjPjj _ _ and let P 1 be the subset of P with one or more tags in E. Since GðxÞ < e for each x outside of E and GðxÞ 1 for all x in [0,1], we get 0 S G; P ¼ S G; P 0 þ S G; P 1 < e þ ð2nÞd ¼ 2e: Since e > 0 is arbitrary, we conclude that G is Riemann integrable with integral equal to & zero. Some Properties of the Integral The difficulties involved in determining the value of the integral and of de suggest that it would be very useful to have some general theorems. The first result in this direction enables us to form certain algebraic combinations of integrable functions. 7.1.5 Theorem Suppose that f and g are in R½a; b. Then: (a) If k 2 R, the function kf is in R½a; b and Z b a Z kf ¼ k b a f: (b) The function f þ g is in R½a; b and Z a b Z ð f þ gÞ ¼ (c) If f ðxÞ gðxÞ for all x 2 ½a; b, then Z b a b a Z f Z fþ b a g: b a g: C07 12/13/2010 10:5:32 Page 205 7.1 RIEMANN INTEGRAL 205 Proof. If P_ ¼ fð½xi1 ; xi ; ti Þgni¼1 is a tagged partition of [a, b], then it is an easy exercise to show that S kf ; P_ ¼ kS f ; P_ ; S f þ g; P_ ¼ S f ; P_ þ S g; P_ ; S f ; P_ S g; P_ : We leave it to the reader to show that the assertion (a) follows from the first equality. As an example, we will complete the proofs of (b) and (c). Given e > 0, we can use the argument in the proof of the Uniqueness Theorem 7.1.2 to _ < de , then both construct a number de > 0 such that if P is any tagged partition with jjPjj Z b Z b S f ; P < e=2 S g; P ð5Þ f and g < e=2: a To prove (b), we note that Z b Z S f þ g; P f þ a a b a Z b Z b g ¼ S f ; P þ S g; P f g a a Z b Z b S f ; P f þ S g; P g a a < e=2 þ e=2 ¼ e: Since e > 0 is arbitrary, we conclude that f þ g 2 R½a; b and that its integral is the sum of the integrals of f and g. To prove (c), we note that the Triangle Inequality applied to (5) implies Z b Z b f e=2 < S f ; P and S g; P < g þ e=2: a a If we use the fact that S f ; P_ S g; P_ , we have Z b Z b f g þ e: a a But, since e > 0 is arbitrary, we conclude that Rb a f Rb a g. Q.E.D. Boundedness Theorem We now show that an unbounded function cannot be Riemann integrable. 7.1.6 Theorem If f 2 R½a; b, then f is bounded on [a, b]. Proof. Assume that f is an unbounded function in R½a; b with integral L. Then there _ < d, then we have that if P_ is any tagged partition of ½a; b with jjPjj exists d > 0 such S f ; P_ L < 1, which implies that S f ; P < jLj þ 1: ð6Þ Now let Q ¼ f½xi1 ; xi gnn¼1 be a partition of [a, b] with jjQjj < d. Since j f j is not bounded on [a, b], then there exists at least one subinterval in Q, say½xk1 ; xk , on which j f j is not bounded—for, if j f j is bounded on each subinterval ½xi1 ; xi by M i, then it is bounded on [a, b] by maxfM 1 ; . . . ; M n g. C07 12/13/2010 10:5:32 206 Page 206 CHAPTER 7 THE RIEMANN INTEGRAL We will now pick tags for Q that will provide a contradiction to (6). We tag Q by tt :¼ xi for i 6¼ k and we pick tk 2 ½xk1 ; xk such that X f ðti Þðxi xi1 Þ: j f ðtk Þðxk xk1 Þj > jLj þ 1 þ i6¼k From the Triangle Inequality (in the form jA þ Bj jAj jBj), we have X S f ; Q j f ðtk Þðxk xk1 Þj f ðti Þðxi xi1 Þ > jLj þ 1; i6¼k which contradicts (6). Q.E.D. We will close this section with an example of a function that is discontinuous at every rational number and is not monotone, but is Riemann integrable nevertheless. 7.1.7 Example We consider Thomae’s function h : ½0; 1 ! R defined, as in Example 5.1.6(h), by hðxÞ :¼ 0 if x 2 ½0; 1 is irrational, hð0Þ :¼ 1 and by hðxÞ :¼ 1=n if x 2 ½0; 1 is the rational number x ¼ m=n where m; n 2 N have no common integer factors except 1. It was seen in 5.1.6(h) that h is continuous at every irrational number and discontinuous at every rational number in [0, l]. See Figure 5.1.2. We will now show that h 2 R½0; 1. For e > 0, the set E :¼ fx 2 ½0; 1 : hðxÞ e=2g is a finite set. (For example, if e=2 ¼ 1=5, then there are eleven values of x such that hðxÞ 1=5, namely, E ¼ f0; 1; 1=2; 1=3; 2=3; 1=4; 3=4; 1=5; 2=5; 3=5; 4=5g. (Sketch a graph.) We let n be the number of elements in E and take d :¼ e=ð4nÞ. If P_ is a given tagged partition such _ < d, then we separate P_ into two subsets. We let P_ 1 be the collection of tagged that jjPjj intervals in P_ that have their tags in E, and we let P_ 2 be the subset of tagged intervals in P_ that have their tags elsewhere in [0, 1]. Allowing for the possibility that a tag of P_ 1 may be an endpoint of adjacent intervals, we see that P_ 1 has at most 2n intervals and the total length of these intervals can be at most 2nd ¼ e=2. Also, we have 0 < hðti Þ 1 for each tag ti in P_ 1 . Consequently, we have S h; P_ 1 1 2nd e=2. For tags ti in P_ 2 , we have _ hðti Þ < e=2 and the total length of the subintervals in P 2 is clearly less than 1, so that S h; P_ 2 < ðe=2Þ 1 ¼ e=2. Therefore, combining these results, we get 0 S h; P ¼ S h; P 1 þ S h; P 2 < e=2 þ e=2 ¼ e: Since e > 0 is arbitrary, we infer that h 2 R½0; 1 with integral 0. & Exercises for Section 7.1 1. If I :¼ ½0; 4, calculate the norms of the following partitions: (a) P 1 :¼ ð0; 1; 2; 4Þ; (b) P 2 :¼ ð0; 2; 3; 4Þ; (c) P 3 :¼ ð0; 1; 1:5; 2; 3:4; 4Þ; (d) P 4 :¼ ð0; :5; 2:5; 3:5; 4Þ: 2. If f ðxÞ :¼ x2 for x 2 ½0; 4, calculate the following Riemann sums, where P_ i has the same partition points as in Exercise 1, and the tags are selected as indicated. (a) P_ 1 with the tags at the left endpoints of the subintervals. (b) P_ 1 with the tags at the right endpoints of the subintervals. (c) P_ 2 with the tags at the left endpoints of the subintervals. (d) P_ 2 with the tags at the right endpoints of the subintervals. 3. Show that f : ½a; b ! R is Riemann integrable on [a, b] if and only if there exists L 2 R such that for every e> 0 there de > 0 such that if P_ is any tagged partition with norm exists _ de , then S f ; P_ L e. jjPjj C07 12/13/2010 10:5:33 Page 207 7.1 RIEMANN INTEGRAL 207 4. Let P_ be a tagged partition of [0, 3]. _ (a) Show that the union U 1 of all subintervals in P_ with tags in [0, 1] satisfies 0; 1 jjPjj _ . U 1 0; 1 þ jjPjj _ (b) Show that the union U 2 of all subintervals in P_ with tags in [1,2] satisfies 1 þ jjPjj; _ 2 þ jjPjj _ : _ U 2 1 jjPjj; 2 jjPjj 5. Let P_ :¼ fðI i ; ti Þgni¼1 be a tagged partition of [a, b] and let c1 < c2 . _ u (a) If u belongs to a subinterval I i whose tag satisfies c1 ti c2 , show that c1 jjPjj _ c2 þ jjPjj. _ v c2 jjPjj, _ then the tag ti of any subinterval I i (b) If v 2 ½a; b and satisfies c1 þ jjPjj that contains v satisfies ti 2 ½c1 ; c2 . 6. (a) Let f ðxÞ :¼ 2 if 0 x < 1 and f ðxÞ :¼ 1 if 1 x 2. Show that f 2 R½0; 2 and evaluate its integral. (b) Let hðxÞ :¼ 2 if 0 x < 1; hð1Þ :¼ 3 and hðxÞ :¼ 1 if 1 < x 2. Show that h 2 R½0; 2 and evaluate its integral. 7. Use Mathematical Induction and Theorem 7.1.5 to show that if f 1 ; . . . ; f n are in R½a; b n X and if k1 ; . . . ; kn 2 R, then the linear combination f ¼ ki f i belongs to R½a; b and Z b n X i¼1 Rb ki f i. a f ¼ a i¼1 R b 8. If f 2 R½a; b and j f ðxÞj M for all x 2 ½a; b, show that a f M ðb aÞ. 9. If f 2 R½a; b and if P_ n is any sequence of tagged partitions of ½a; b such that jjP_ n jj ! 0, Rb prove that f ¼ limn S f ; P_ n . a 10. Let gðxÞ :¼ 0 if x 2 ½0; 1 is rational and gðxÞ :¼ 1=x if x 2 ½0; 1 is irrational. Explain why g 2 = R½0; 1. However, show exists a sequence P_ n of tagged partitions of [a, b] such that there _ _ that jjP n jj ! 0 and limn S g; P n exists. 11. Suppose that f is bounded on [a, b] and that there exists two sequences of of tagged partitions _ n jj ! 0, but such that limn S f ; P_ n 6¼ limn S f ; Q _ n . Show [a, b] such that jjP_ n jj ! 0 and jjQ that f is not in R½a; b. 12. Consider the Dirichlet function, introduced in Example 5.1.6(g), defined by f ðxÞ :¼ 1 for x 2 ½0; 1 rational and f ðxÞ :¼ 0 for x 2 ½0; 1 irrational. Use the preceding exercise to show that f is not Riemann integrable on [0, 1]. 13. Suppose that c d are points in [a, b]. If w : ½a; b ! R satisfies R bwðxÞ ¼ a > 0 for x 2 ½c; d and wðxÞ ¼ 0 elsewhere in [a, b], prove that w 2 R½a; b and that a w ¼ aðd cÞ. [Hint: Given _ < de then we have aðd c 2de Þ S w; P_ e > 0 let de :¼ e=4a and show that if jjPjj aðd c þ 2de Þ: 14. Let 0 a < b; let QðxÞ :¼ x2 for x 2 ½a; b and let P :¼ f½xi1 ; xi gni¼1 be a partition of [a, b]. For each i, let qi be the positive square root of 2 1 2 3 xi þ xi xi1 þ xi1 : (a) Show that qi satisfies 0 xi1 qi xi : (b) Show that Qðqi Þðxi xi1 Þ ¼ 13 x3i x3i1 . _ is the tagged partition with the same subintervals as P and the tags q , show that (c) If Q i _ ¼ 1 b3 a3 . S Q; Q 3 (d) Use the argument in Example 7.1.4(c) to show that Q 2 R½a; b and Z b Z b Q¼ x2 dx ¼ 13 b3 a3 : a a 15. If f 2 R½a; b and c 2 R, we define g on [a þ c; b þ c] by gðyÞ :¼ f ðy cÞ. Prove that Rb R bþc g 2 R½a þ c; b þ c and that aþc g ¼ a f . The function g is called the c-translate of f. C07 12/13/2010 10:5:33 208 Page 208 CHAPTER 7 THE RIEMANN INTEGRAL Section 7.2 Riemann Integrable Functions We begin with a proof of the important Cauchy Criterion. We will then prove the Squeeze Theorem, which will be used to establish the Riemann integrability of several classes of functions (step functions, continuous functions, and monotone functions). Finally we will establish the Additivity Theorem. We have already noted that direct use of the definition requires that we know the value of the integral. The Cauchy Criterion removes this need, but at the cost of considering two Riemann sums, instead of just one. 7.2.1 Cauchy Criterion A function: ½a; b ! R belongs to R½a; b if and only if for _ are any tagged partitions of [a, b] with every e > 0 there exists he > 0 such that if P_ and Q _ < he , then _ < he and jjQjj jjPjj S f ; P S f ; Q < e: & _ are _ Q Proof. ð)Þ If f 2 R½a; b with integral L, let he :¼ de=2 > 0 be such that if P, _ < he , then _ < he and jjQjj tagged partitions such that jjPjj S f ; P L < e=2 and S f ; Q L < e=2: Therefore we have S f ; P S f ; Q S f ; P L þ L S f ; Q S f ; P L þ L S f ; Q < e=2 þ e=2 ¼ e: _ are tagged partitions with ð(Þ For each n 2 N, let dn > 0 be such that if P_ and Q norms < dn , then S f ; P f f ; Q < 1=n: Evidently we may assume that dn dnþ1 for n 2 N; otherwise, we replace dn by d0n :¼ minfd1 ; . . . ; dn g. For each n 2 N, let P_ n be a tagged partition with jjP_ n jj < dn . Clearly, if m > n then both P_ m and P_ n have norms < dn , so that S f ; P n S f ; P m < 1=n ð1Þ for m > n: 1 Consequently, the sequence S f ; P_ m m¼1 is a Cauchy sequence in R. Therefore (by Theorem 3.5.5) this sequence converges in R and we let A :¼ limm S f ; P_ m . Passing to the limit in (1) as m ! 1, we have S f ; P n A 1=n for all n 2 N: _ is To see that A is the Riemann integral of f, given e > 0, let K 2 N satisfy K > 2=e. If Q _ any tagged partition with jjQjj < dK , then S f ; Q A S f ; Q S f ; P K þ S f ; P K A 1=K þ 1=K < e: Since e > 0 is arbitrary, then f 2 R½a; b with integral A. We will now give two examples of the use of the Cauchy Criterion. Q.E.D. C07 12/13/2010 10:5:34 Page 209 7.2 RIEMANN INTEGRABLE FUNCTIONS 209 7.2.2 Examples (a) Let g : ½0; 3 ! R be the function considered in Example 7.1.4(b). _ < d, then In that example we saw that if P is a tagged partition of [0, 3] with norm jjPjj 8 5d S g; P 8 þ 5d: _ is another tagged partition with jjQjj _ < d, then Hence if Q 8 5d S g; Q 8 þ 5d: If we subtract these two inequalities, we obtain S g; P S g; Q 10d: In order to make this final term < e, we are led to employ the Cauchy Criterion with he :¼ e=20. (We leave the details to the reader.) (b) The Cauchy Criterion can be used to show that a function f : ½a; b ! R is not Riemann integrable. To do this we need to show that: There exists e0 > 0 such that for any _ with jjPjj _ < h such that _ < h and jjQjj h partitions P_ and Q > 0 there existstagged _ e0 . S f ; P_ S f ; Q We will apply these remarks to the Dirichlet function, considered in 5.1.6(g), defined by f ðxÞ :¼ 1 if x 2 ½0; 1 is rational and f ðxÞ :¼ 0 if x 2 ½0; 1 is irrational. 1 _ Here we take e0 :¼ 2. If P is any partition all of whose tags are rational numbers then _ is any tagged partition all of whose tags are irrational numbers S f ; P_ ¼ 1, while if Q _ ¼ 0. Since we are able to take such tagged partitions with arbitrarily small then S f ; Q & norms, we conclude that the Dirichlet function is not Riemann integrable. The Squeeze Theorem In working with the definition of Riemann integral, we have encountered two types of difficulties. First, for each partition, there are infinitely many choices of tags. And second, there are infinitely many partitions that have a norm less than a specified amount. We have experienced dealing with these difficulties in examples and proofs of theorems. We will now establish an important tool for proving integrability called the Squeeze Theorem that will provide some relief from those difficulties. It states that if a given function can be ‘‘squeezed’’ or bracketed between two functions that are known to be Riemann integrable with sufficient accuracy, then we may conclude that the given function is also Riemann integrable. The exact conditions are given in the statement of the theorem. (The idea of squeezing a function to establish integrability led the French mathematician Gaston Darboux to develop an approach to integration by means of upper and lower integrals, and this approach is presented in Section 7.4.) 7.2.3 Squeeze Theorem Let f : ½a; b ! R. Then f 2 R½a; b if and only if for every e > 0 there exist functions ae and ve in R½a; b with ð2Þ and such that ð3Þ ae ðxÞ f ðxÞ ve ðxÞ Z a b for all ðve ae Þ < e: x 2 ½a; b; C07 12/13/2010 10:5:34 210 Page 210 CHAPTER 7 THE RIEMANN INTEGRAL Proof. ()) Take ae ¼ ve ¼ f for all e > 0. (() Let e > 0. Since ae and ve belong to R½a; b, there exists de > 0 such that if P_ is _ < de then any tagged partition with jjPjj Z b Z b S ae ; P and ae < e ve < e: S v e ; P a It follows from these inequalities that Z b ae e < S ae ; P a S ve ; P < and a Z b a ve þ e: In view of inequality (2), we have S ae ; P_ S f ; P_ S ve ; P_ , whence Z b Z b ae e < S f ; P < ve þ e: a a _ is another tagged partition with jjQjj _ < de , then we also have If Q Z b Z b ae e < S f ; Q < ve þ e: a a If we subtract these two inequalities and use (3), we conclude that Z b Z b S f ; P S f ; Q < ve ae þ 2e a Z ¼ a a b ðve ae Þ þ 2e < 3e: Since e > 0 is arbitrary, the Cauchy Criterion implies that f 2 R½a; b. Q.E.D. Classes of Riemann Integrable Functions The Squeeze Theorem is often used in connection with the class of step functions. It will be recalled from Definition 5.4.9 that a function w : ½a; b ! R is a step function if it has only a finite number of distinct values, each value being assumed on one or more subintervals of [a, b]. For illustrations of step functions, see Figures 5.4.3 or 7.1.4. 7.2.4 Lemma If J is a subinterval of [a, b] having endpoints c < Rd and if wJ ðxÞ :¼ 1 for b x 2 J and wJ ðxÞ :¼ 0 elsewhere in [a, b], then wJ 2 R½a; b and a wJ ¼ d c. Proof. If J ¼ ½c; d with c d, this is Exercise 7.1.13 and we can choose de :¼ e=4. There are three other subintervals J having the same endpoints c and d, namely, [c, d ), (c, d ], and (c, d ). Since, by Theorem 7.1.3, we can change the value of a function at finitely many points without changing the integral, we have the same result for these other three subintervals. Therefore, we conclude that all four functions wJ are integrable with integral equal to d c. Q.E.D. It is an important fact that any step function is Riemann integrable. 7.2.5 Theorem If w : ½a; b ! R is a step function, then w 2 R½a; b. Proof. Step functions of the type appearing in 7.2.4 are called ‘‘elementary step functions.’’ In Exercise 5 it is shown that an arbitrary step function w can be expressed C07 12/13/2010 10:5:34 Page 211 7.2 RIEMANN INTEGRABLE FUNCTIONS 211 as a linear combination of such elementary step functions: ð4Þ w¼ m X j¼1 k j wJ j ; where J j has endpoints cj < d j . The lemma and Theorem 7.1.5(a, b) imply that w 2 R½a; b and that Z b m X Q.E.D. ð5Þ w¼ k j d j cj : a j¼1 We illustrate the use of step functions and the Squeeze Theorem in the next two examples. The first reconsiders a function that originally required a complicated calculation. 7.2.6 Examples (a) The function g in Example 7.1.4(b) is defined by gðxÞ ¼ 2 for 0 x 1 and gðxÞ ¼ 3 for 1 <R x 3. We now see that g is a step function and therefore 3 we calculate its integral to be 0 g ¼ 2 ð1 0Þ þ 3 ð3 1Þ ¼ 2 þ 6 ¼ 8. (b) Let hðxÞ :¼ x on [0,1] and let Pn :¼ ð0; 1=n; 2=n; . . . ; ðn 1Þ=n; n=n ¼ 1Þ. We define the step functions an and vn on the disjoint subintervals ½0; 1=nÞ; ½1=n; 2=nÞ; . . . ; ½ðn 2Þ=nÞ; ðn 1Þ=nÞ; ½ðn 1Þ=n; 1 as follows: an ðxÞ :¼ hððk 1Þ=nÞ ¼ ðk 1Þ=n for x in ½ðk 1Þ=n; k=nÞ for k ¼ 1; 2; . . . ; n 1, and an ðxÞ :¼ hððn 1Þ=nÞ ¼ ðn 1Þ=n for x in ½ðn 1Þ=n; 1. That is, an has the minimum value of h on each subinterval. Similarly, we define vn to be the maximum value of h on each subinterval, that is, vn ðxÞ :¼ k=n for x in ½ðk 1Þ=n; k=nÞ for k ¼ 1; 2; . . . ; n 1, and vn ðxÞ :¼ 1 for x in ½ðn 1Þ=n; 1. (The reader should draw a sketch for the case n ¼ 4.) Then we get Z 1 1 an ¼ ð0 þ 1=n þ 2=n þ þ ðn 1Þ=nÞ n 0 1 ¼ 2 ð 1 þ 2 þ þ ð n 1Þ Þ n 1 ðn 1Þn 1 ¼ ð1 1=nÞ: ¼ 2 n 2 2 R1 In a similar manner, we also get 0 vn ¼ 12 ð1 þ 1=nÞ. Thus we have an ðxÞ hðxÞ vn ðxÞ for x 2 ½0; 1 and Z 0 1 1 ðvn an Þ ¼ : n Since for a given e > 0, we can choose n so that 1n < e, it follows from the Squeeze Theorem that h is integrable. We also see that the value of the integral of h lies between the integrals & of an and vn for all n and therefore has value 12. We will now use the Squeeze Theorem to show that an arbitrary continuous function is Riemann integrable. C07 12/13/2010 10:5:35 212 Page 212 CHAPTER 7 THE RIEMANN INTEGRAL 7.2.7 Theorem If f : ½a; b ! R is continuous on [a, b], then f 2 R½a; b. Proof. It follows from Theorem 5.4.3 that f is uniformly continuous on [a, b]. Therefore, given e > 0 there exists de > 0 such that if u; v 2 ½a; b and ju vj < de , then we have j f ðuÞ f ðvÞj < e=ðb aÞ. Let P ¼ fI i gni¼1 be a partition such that jjPjj < de . Applying Theorem 5.3.4 we let ui 2 I i be a point where f attains its minimum value on Ii, and let vi 2 I i be a point where f attains its maximum value on I i . Let ae be the step function defined by ae ðxÞ :¼ f ðui Þ for x 2 ½xi1 ; xi Þ ði ¼ 1; . . . ; n 1Þ and ae ðxÞ :¼ f ðun Þ for x 2 ½xn1 ; xn . Let ve be defined similarly using the points vi instead of the ui . Then one has ae ðxÞ f ðxÞ ve ðxÞ for all x 2 ½a; b: Moreover, it is clear that Z 0 b a ðv e a e Þ ¼ n X ð f ðvi Þ f ðui ÞÞðxi xi1 Þ i¼1 < n X i¼1 e ðxi xi1 Þ ¼ e: ba Therefore it follows from the Squeeze Theorem that f 2 R½a; b. Q.E.D. Monotone functions are not necessarily continuous at every point, but they are also Riemann integrable. 7.2.8 Theorem If f : ½a; b ! R is monotone on [a, b], then f 2 R½a; b. Proof. Assume that f is increasing on I ¼ ½a; b. Partitioning the interval into n equal subintervals I k ¼ ½xk1 ; xk gives us xk xk1 ¼ ðb aÞ=n; k ¼ 1; 2; . . . ; n. Since f is increasing on I k , its minimum value is attained at the left endpoint xk1 and its maximum value is attained at the right endpoint xk . Therefore, we define the step functions aðxÞ :¼ f ðxk1 Þ and vðxÞ :¼ f ðxk Þ for x 2 ½xk1 ; xk Þ; k ¼ 1; 2; . . . ; n 1, and aðxÞ :¼ f ðxn1 Þ and vðxÞ :¼ f ðxn Þ for x 2 ½xn1 ; xn . Then we have aðxÞ f ðxÞ vðxÞ for all x 2 I, and Z b a Z a b a ¼ ba ð f ðx0 Þ þ f ðx1 Þ þ þ f ðxn1 ÞÞ n v¼ ba ð f ðx1 Þ þ þ f ðxn1 Þ þ f ðxn ÞÞ: n Subtracting, and noting the many cancellations, we obtain Z a b ðv a Þ ¼ ba ba ð f ðbÞ f ðaÞÞ: ð f ðxn Þ f ðx0 ÞÞ ¼ n n Thus R b for a given e > 0, we choose n such that n > ðb aÞð f ðbÞ f ðaÞÞ=e. Then we have Q.E.D. a ðv aÞ < e and the Squeeze Theorem implies that f is integrable on I. C07 12/13/2010 10:5:35 Page 213 7.2 RIEMANN INTEGRABLE FUNCTIONS 213 The Additivity Theorem We now return to arbitrary Riemann integrable functions. Our next result shows that the integral is an ‘‘additive function’’ of the interval over which the function is integrated. This property is no surprise, but its proof is a bit delicate and may be omitted on a first reading. 7.2.9 Additivity Theorem Let f :¼ ½a; b ! R and let c 2 ða; bÞ. Then f 2 R½a; b if and only if its restrictions to [a, c] and [c, b] are both Riemann integrable. In this case Z b Z c Z b ð6Þ f ¼ fþ f: a a c Proof. ð(Þ Suppose that the restriction f1 of f to [a, c], and the restriction f2 of f to [c, b] are Riemann integrable to L1 and L2 respectively. Then, given e > 0 there exists d0 > 0 such that if P_ 1 is a tagged partition of [a, c] with jjP_ 1 jj < d0 , then S f 1 ; P_ 1 L1 < e=3. Also there exists d00 > 0 such that if P_ 2 is a tagged partition of [c, b] wilh jjP_ 2 jj < d00 then S f 2 ; P_ 2 L2 < e=3. If M is a bound for j f j, we define de :¼ minfd0 ; d00 ; e=6M g and let _ < d. We will prove that P_ be a tagged partition of [a, b] with jjQjj ð7Þ S f ; Q ðL1 þ L2 Þ < e: _ _ _ (i) If c is a partition point of Q, we split Q into a partition Q1 of [a, c] and a 0partition _ _ _ _2 _ _ Q2 of [c, b]. Since S f ; Q ¼ S f ; Q1 þ S f ; Q2 , and since Q1 has norm < d and Q 00 has norm < d , the inequality (7) is clear. _ ¼ fðI k ; tk Þgm , there exists k m such that (ii) If c is not a partition point in Q k¼1 _ c 2 ðxk1 ; xk Þ. We let Q1 be the tagged partition of [a, c] defined by Q1 :¼ fðI 1 ; t1 Þ; . . . ; ðI k1 ; tk1 Þ; ð½xk1 ; c; cÞg; _ 2 be the tagged partition of [c, b] defined by and Q Q2 :¼ fð½c; xk ; cÞ; ðI kþ1 ; tkþ1 Þ; . . . ; ðI m ; tm Þg: A straightforward calculation shows that S f ; Q S f ; Q1 S f ; Q2 ¼ f ðtk Þðxk xk1 Þ f ðcÞðxk xk1 Þ ¼ ð f ðtk Þ f ðcÞÞ ðxk xk1 Þ; whence it follows that S f ; Q S f ; Q 2M ðxk xk1 Þ < e=3: 1 S f ; Q2 _ 1 jj < d d0 and jjQ _ 2 jj < d d00 , it follows that But since jjQ S f ; Q1 L1 < e=3 S f ; Q2 L2 < e=3; and from which we obtain (7). Since e > 0 is arbitrary, we infer that f 2 R½a; b and that (6) holds. ð)Þ We suppose that f 2 R½a; b and, given e > 0, we let he > 0 satisfy the Cauchy _ 1 be tagged partitions of Criterion 7.2.1. Let f1 be the restriction of f to [a, c] and let P_ 1 ; Q C07 12/13/2010 10:5:36 214 Page 214 CHAPTER 7 THE RIEMANN INTEGRAL _ 1 jj < he . By adding additional partition points and tags from [a, c] with jjP_ 1 jj < he and jjQ _ 1 to tagged partitions P_ and Q _ of [a, b] that satisfy jjPjj _ _ < he [c, b], we can extend P 1 and Q _ _ then _ and jjQjj < he . If we use the same additional points and tags in [c, b] for both P and Q, S f 1 ; P 1 S f 1 ; Q1 ¼ S f ; P S f ; Q : _ 1 < e. Therefore the Cauchy _ have norm he , then S f ; P_ 1 S f ; Q Since both P_ and Q 1 1 Condition shows that the restriction f1 of f to [a, c] is in R½a; c. In the same way, we see that the restriction f2 of f to [c, b] is in R½c; d . The equality (6) now follows from the first part of the theorem. Q.E.D. 7.2.10 Corollary If f 2 R½a; b , and if ½c; d ½a; b , then the restriction of f to [c, d ] is in R½c; d . Proof. Since f 2 R½a; b and c 2 ½a; b, it follows from the theorem that its restriction to [c, b] is in R½c; b. But if d 2 ½c; b, then another application of the theorem shows that the restriction of f to ½c; d is in R½c; d . Q.E.D. 7.2.11 Corollary If f 2 R½a; b and if a ¼ c0 < c1 < < cm ¼ b, then the restrictions of f to each of the subintervals ½ci1 ; ci are Riemann integrable and Z b m Z ci X f ¼ f: a i¼1 ci1 Until now, we have considered the Riemann integral over an interval [a, b] where a < b. It is convenient to have the integral defined more generally. 7.2.12 Definition If f 2 R½a; b and if a; b 2 ½a; b with a < b, we define Z a Z b Z a f :¼ f and f :¼ 0: b a a 7.2.13 Theorem If f 2 R½a; b and if a; b; g are any numbers in [a, b], then Z b Z g Z b ð8Þ f ¼ fþ f; a a g in the sense that the existence of any two of these integrals implies the existence of the third integral and the equality (8). Proof. If any two of the numbers a; b; g are equal, then (8) holds. Thus we may suppose that all three of these numbers are distinct. For the sake of symmetry, we introduce the expression Z b Z g Z a Lða; b; g Þ :¼ fþ fþ f: a b g It is clear that (8) holds if and only if Lða; b; g Þ ¼ 0. Therefore, to establish the assertion, we need to show that L ¼ 0 for all six permutations of the arguments a; b, and g. We note that the Additivity Theorem 7.2.9 implies that Lða; b; g Þ ¼ 0 when a < g < b. But it is easily seen that both Lðb; g; aÞ and Lðg; a; bÞ equal Lða; b; g Þ. C07 12/13/2010 10:5:36 Page 215 7.2 RIEMANN INTEGRABLE FUNCTIONS 215 Moreover, the numbers Lðb; a; g Þ; Lða; g; bÞ; and Lðg; b; aÞ are all equal to Lða; b; g Þ. Therefore, L vanishes for all possible configurations of these three points. Q.E.D. Exercises for Section 7.2 1. Let f : ½a; b ! R. Show that f 2 = R½a; b if and only if there exists e0 > 0 such that for every _ _ _ _ n 2 N there exist tagged partitions P n and Qn with jjP n jj < 1=n and jjQn jj < 1=n such that _ n e0 . S f ; P_ n S f ; Q 2. Consider the function h defined by hðxÞ :¼ x þ 1 for x 2 ½0; 1 rational, and hðxÞ :¼ 0 for x 2 ½0; 1 irrational. Show that h is not Riemann integrable. 3. Let H ðxÞ :¼ k for x ¼ 1=k ðk 2 N Þ and H ðxÞ :¼ 0 elsewhere on [0, 1]. Use Exercise 1, or the argument in 7.2.2(b), to show that H is not Riemann integrable. 4. If aðxÞ :¼ x and vðxÞ :¼ x and if aðxÞ f ðxÞ vðxÞ for all x 2 ½0; 1, does it follow from the Squeeze Theorem 7.2.3 that f 2 R½0; 1? 5. If J is any subinterval of [a, b] and if wJ ðxÞ :¼ 1 for x 2 J and wJ ðxÞ :¼ 0 elsewhere on [a, b], we say that wJ is an elementary step function on [a, b]. Show that every step function is a linear combination of elementary step functions. 6. If c : ½a; b ! R takes on only a finite number of distinct values, is c a step function? 7. If S f ; P_ is any Riemann R b sum of f : ½a; b ! R, show that there exists a step function w : ½a; b ! R such that a w ¼ S f ; P_ . Rb 8. Suppose that f is continuous on [a, b], that f ðxÞ 0 for all x 2 ½a; b and that a f ¼ 0. Prove that f ðxÞ ¼ 0 for all x 2 ½a; b. 9. Show that the continuity hypothesis in the preceding exercise cannot be dropped. Rb Rb 10. If f and g are continuous on [a, b] and if a f ¼ a g, prove that there exists c 2 ½a; b such that f ðcÞ ¼ gðcÞ. 11. If f is bounded by M on [a, b] and if the restriction of f to every interval [c, b] where c 2 ða; bÞ is Rb Rb Riemann integrable, show that f 2 R½a; b and that c f ! a f as c ! aþ. [Hint: Let ac ðxÞ :¼ M and vc ðxÞ :¼ M for x 2 ½a; cÞ and ac ðxÞ :¼ vc ðxÞ :¼ f ðxÞ for x 2 ½c; b. Apply the Squeeze Theorem 7.2.3 for c sufficiently near a.] 12. Show that gðxÞ :¼ sinð1=xÞ for x 2 ð0; 1 and gð0Þ :¼ 0 belongs to R½0; 1. 13. Give an example of a function f : ½a; b ! R that is in R½c; b for every c 2 ða; bÞ but which is not in R½a; b. 14. Suppose that f : ½a; b ! R, that a ¼ c0 < c1 < < cm ¼ b and that the restrictions of f to ½ci1 ; ci belong to R½ci1 ; ci for i ¼ 1; . . . ; m. Prove that f 2 R½a; b and that the formula in Corollary 7.2.11 holds. 15. If f is bounded and there is a finite set E such that f is continuous at every point of ½a; bnE, show that f 2 R½a; b. Rb 16. If f is continuous on [a, b], a < b, show that there exists c 2 ½a; b such that we have a f ¼ f ðcÞðb aÞ. This result is sometimes called the Mean Value Theorem for Integrals. 17. If f and g are [a, b] and gðxÞ > 0 for all x 2 ½a; b, show that there exists c 2 ½a; b R b continuousR on b such that a f g ¼ f ðcÞ a g. Show that this conclusion fails if we do not have gðxÞ > 0. (Note that this result is an extension of the preceding exercise.) C07 12/13/2010 10:5:36 216 Page 216 CHAPTER 7 THE RIEMANN INTEGRAL 18. Let f be continuous on [a, b], let f ðxÞ 0 for x 2 ½a; b, and let M n :¼ ð limðM n Þ ¼ supf f ðxÞ : x 2 ½a; bg. Rb a f n Þ1=n . Show that 19. Suppose that a > 0 and that f 2 R½a; a. Ra Ra (a) If f is even (that is, if f ðxÞ ¼ f ðxÞ for all x 2 ½0; aÞ, show that a f ¼ 2 0 f . Ra (b) If f is odd (that is, if f ðxÞ ¼ f ðxÞ for all x 2 ½0; aÞ, show that a f ¼ 0. Ra Ra 20. If f is continuous on ½a; a, show that a f ðx2 Þdx ¼ 2 0 f ðx2 Þdx. Section 7.3 The Fundamental Theorem We will now explore the connection between the notions of the derivative and the integral. In fact, there are two theorems relating to this problem: one has to do with integrating a derivative, and the other with differentiating an integral. These theorems, taken together, are called the Fundamental Theorem of Calculus. Roughly stated, they imply that the operations of differentiation and integration are inverse to each other. However, there are some subtleties that should not be overlooked. The Fundamental Theorem (First Form) The First Form of the Fundamental Theorem provides a theoretical basis for the method of calculating an integral that the reader learned in calculus. It asserts that if a function R b f is the derivative of a function F, and if f belongs to R ½ a; b , then the integral a f can be calculated by means of the evaluation F ba :¼ F ðbÞ F ðaÞ. A function F such that F 0 ðxÞ ¼ f ðxÞ for all x 2 ½a; b is called an antiderivative or a primitive of f on [a, b]. Thus, when f has an antiderivative, it is a very simple matter to calculate its integral. In practice, it is convenient to allow some exceptional points c where F 0 ðcÞ does not exist in R, or where it does not equal f ðcÞ. It turns out that we can permit a finite number of such exceptional points. 7.3.1 Fundamental Theorem of Calculus (First Form) Suppose there is a finite set E in [a, b] and functions f, F :¼ ½a; b ! R such that: (a) F is continuous on [a, b], (b) F 0 ðxÞ ¼ f ðxÞ for all x 2 ½a; bnE, (c) f belongs to R½a; b. Then we have ð1Þ Z b a f ¼ F ðbÞ F ðaÞ: Proof. We will prove the theorem in the case where E :¼ fa; bg. The general case can be obtained by breaking the interval into the union of a finite number of intervals (see Exercise 1). Let e > 0 be given. Since f 2 R½a; b by assumption (c), there exists de > 0 such that if _ < de , then P_ is any tagged partition with jjPjj Z b ð2Þ f < e: S f ; P a C07 12/13/2010 10:5:37 Page 217 7.3 THE FUNDAMENTAL THEOREM 217 If the subintervals in P_ are ½xi1 ; xi , then the Mean Value Theorem 6.2.4 applied to F on ½xi1 ; xi implies that there exists ui 2 ðxi1 ; xi Þ such that F ðxi Þ F ðxi1 Þ ¼ F 0 ðui Þ ðxi xi1 Þ for i ¼ 1; . . . ; n: If we add these terms, note the telescoping of the sum, and use the fact that F 0 ðui Þ ¼ f ðui Þ, we obtain F ð bÞ F ð aÞ ¼ n X ðF ðxi Þ F ðxi1 ÞÞ ¼ i¼1 n X f ðui Þðxi xi1 Þ: i¼1 Now let P_ u :¼ fð½xi1 ; xi ; ui Þgni¼1 , so the sum on the right equals S f ; P_ u . If we substitute F ðbÞ F ðaÞ ¼ S f ; P_ u into (2), we conclude that Z b F ð bÞ F ð aÞ f < e: a But, since e > 0 is arbitrary, we infer that equation (1) holds. Q.E.D. Remark If the function F is differentiable at every point of [a, b], then (by Theorem 6.1.2) hypothesis (a) is automatically satisfied. If f is not defined for some point c 2 E, we take f ðcÞ :¼ 0. Even if F is differentiable at every point of [a, b], condition (c) is not automatically satisfied, since there exist functions F such that F 0 is not Riemann integrable. (See Example 7.3.2(e).) 7.3.2 Examples (a) If F ðxÞ :¼ 12 x2 for all x 2 ½a; b, then F 0 ðxÞ ¼ x for all x 2 ½a; b. Further, f ¼ F 0 is continuous so it is in R½a; b. Therefore the Fundamental Theorem (with E ¼ ;) implies that Z b x dx ¼ F ðbÞ F ðaÞ ¼ 12 b2 a2 : a (b) If GðxÞ :¼ Arctan x for x 2 ½a; b, then G0 ðxÞ ¼ 1=ðx2 þ 1Þ for all x 2 ½a; b; also G0 is continuous, so it is in R½a; b. Therefore the Fundamental Theorem (with E ¼ ;) implies that Z b 1 dx ¼ Arctan b Arctan a: 2þ1 x a (c) If AðxÞ :¼ jxj for x 2 ½10; 10, then A0 ðxÞ ¼ 1 if x 2 ½10; 0Þ and A0 ðxÞ ¼ þ1 for x 2 ð0; 10. Recalling the definition of the signum function (in 4.1.10(b)), we have A0 ðxÞ ¼ sgnðxÞ for all x 2 ½10; 10nf0g. Since the signum function is a step function, it belongs to R½10; 10. Therefore the Fundamental Theorem (with E ¼ f0g) implies that Z 10 sgnðxÞ dx ¼ Að10Þ Að10Þ ¼ 10 10 ¼ 0: 10 pffiffiffi pffiffiffi (d) If H ðxÞ :¼ 2 x for x 2 ½0; b, then H is continuous on [0, b] and H 0 ðxÞ ¼ 1= x for x 2 ð0; b. Since h :¼ H 0 is not bounded on (0, b], it does not belong to R½0; b no matter how we define hð0Þ. Therefore, the Fundamental Theorem 7.3.1 does not apply. (However, we will see in Example 10.1.10(a) that h is generalized Riemann integrable on [0, b].) C07 12/13/2010 10:5:37 218 Page 218 CHAPTER 7 THE RIEMANN INTEGRAL (e) Let K ðxÞ :¼ x2 cosð1=x2 Þ for x 2 ð0; 1 and let K ð0Þ :¼ 0. It follows from the Product Rule 6.1.3(c) and the Chain Rule 6.1.6 that K 0 ðxÞ ¼ 2x cosð1=x2 Þ þ ð2=xÞsinð1=x2 Þ x 2 ð0; 1: for Further, as in Example 6.1.7(e), it can be shown that K 0 ð0Þ ¼ 0. Thus K is continuous and differentiable at every point of [0, 1]. Since it can be seen that the function K 0 is not bounded on [0, 1], it does not belong to R½0; 1 and the Fundamental Theorem 7.3.1 does not apply to K 0 . (However, we will see from Theorem 10.1.9 that K 0 is generalized Riemann & integrable on [0, 1].) The Fundamental Theorem (Second Form) We now turn to the Fundamental Theorem (Second Form) in which we wish to differentiate an integral involving a variable upper limit. 7.3.3 Definition If f 2 R½a; b, then the function defined by Z z ð3Þ F ðzÞ :¼ f for z 2 ½a; b; a is called the indefinite integral of f with basepoint a. (Sometimes a point other than a is used as a basepoint; see Exercise 6.) We will first show that if f 2 R½a; b, then its indefinite integral F satisfies a Lipschitz condition; hence F is continuous on [a, b]. 7.3.4 Theorem The indefinite integral F defined by (3) is continuous on [a, b]. In fact, if j f ðxÞj M for all x 2 ½a; b, then jF ðzÞ F ðw Þj M jz w j for all z; w 2 ½a; b. Proof. The Additivity Theorem 7.2.9 implies that if z; w 2 ½a; b and w z, then Z z Z w Z z Z z F ð zÞ ¼ f ¼ fþ f ¼ F ðw Þ þ f; a a w whence we have w Z F ð zÞ F ð w Þ ¼ w z f: Now if M f ðxÞ M for all x 2 ½a; b, then Theorem 7.1.5(c) implies that Z z M ðz w Þ f M ðz w Þ; w whence it follows that Z jF ðzÞ F ðw Þj w as asserted. z f M jz w j; Q.E.D. We will now show that the indefinite integral F is differentiable at any point where f is continuous. C07 12/13/2010 10:5:37 Page 219 7.3 THE FUNDAMENTAL THEOREM 219 7.3.5 Fundamental Theorem of Calculus (Second Form) Let f 2 R½a; b and let f be continuous at a point c 2 ½a; b . Then the indefinite integral, defined by (3), is differentiable at c and F 0 ðcÞ ¼ f ðcÞ. Proof. We will suppose that c 2 ½a; bÞ and consider the right-hand derivative of F at c. Since f is continuous at c, given e > 0 there exists he > 0 such that if c x < c þ he , then ð4Þ f ðcÞ e < f ðxÞ < f ðcÞ þ e: Let h satisfy 0 < h < he. The Additivity Theorem 7.2.9 implies that f is integrable on the intervals ½a; c; ½a; c þ h and ½c; c þ h and that Z cþh f: F ðc þ hÞ F ðcÞ ¼ c Now on the interval ½c; c þ h the function f satisfies inequality (4), so that we have Z cþh f ð f ðcÞ þ eÞ h: ð f ðcÞ eÞ h F ðc þ hÞ F ðcÞ ¼ c If we divide by h > 0 and subtract f ðcÞ, we obtain F ð c þ hÞ F ð c Þ f ðcÞ e: h But, since e > 0 is arbitrary, we conclude that the right-hand limit is given by lim x!0þ F ð c þ hÞ F ð c Þ ¼ f ðcÞ: h It is proved in the same way that the left-hand limit of this difference quotient also equals f ðcÞ when c 2 ða; b, whence the assertion follows. Q.E.D. If f is continuous on all of [a, b], we obtain the following result. 7.3.6 Theorem If f is continuous on [a, b], then the indefinite integral F, defined by (3), is differentiable on [a, b] and F 0 ðxÞ ¼ f ðxÞ for all x 2 ½a; b. Theorem 7.3.6 can be summarized: If f is continuous on [a, b], then its indefinite integral is an antiderivative of f. We will now see that, in general, the indefinite integral need not be an antiderivative (either because the derivative of the indefinite integral does not exist or does not equal f ðxÞ). 7.3.7 Examples (a) If f ðxÞ :¼ sgn x on ½1; 1, then f 2 R½1; 1 and has the indefinite integral F ðxÞ :¼ jxj 1 with the basepoint 1. However, since F 0 ð0Þ does not exist, F is not an antiderivative of f on ½1; 1. (b) If h Rdenotes Thomae’s function, considered in 7.1.7, then its indefinite integral x H ðxÞ :¼ 0 h is identically 0 on [0, 1]. Here, the derivative of this indefinite integral exists at every point and H 0 ðxÞ ¼ 0. But H 0 ðxÞ 6¼ hðxÞ whenever x 2 Q \ ½0; 1, so that H is not & an antiderivative of h on [0, 1]. C07 12/13/2010 10:5:37 220 Page 220 CHAPTER 7 THE RIEMANN INTEGRAL Substitution Theorem The next theorem provides the justification for the ‘‘change of variable’’ method that is often used to evaluate integrals. This theorem is employed (usually implicitly) in the evaluation by means of procedures that involve the manipulation of ‘‘differentials,’’ common in elementary courses. 7.3.8 Substitution Theorem Let J :¼ ½a; b and let w : J ! R have a continuous derivative on J. If f : I ! R is continuous on an interval I containing wðJ Þ, then Z b Z wðbÞ ð5Þ f ðwðtÞÞ w0 ðtÞdt ¼ f ðxÞdx: a wðaÞ The proof of this theorem is based on the Chain Rule 6.1.6, and will be outlined in Exercise 17. The hypotheses that f and w0 are continuous are restrictive, but are used to ensure the existence of the Riemann integral on the left side of (5). Z 4 pffiffi sin t pffiffi dt. 7.3.9 Examples (a) Consider the integral t 1 pffiffi pffiffi Here we substitute wðtÞ :¼ t for t 2 ½1; 4 so that w0 ðtÞ ¼ 1= 2 t is continuous on [1, 4]. If we let f ðxÞ :¼ 2 sin x, then the integrand has theR form ð f wÞ w0 and the 2 Substitution Theorem 7.3.8 implies that the integral equals 1 2 sin x dx ¼ 2 cos xj21 ¼ 2ðcos 1 cos 2Þ. Z 4 pffiffi sin t pffiffi dt. (b) Consider the integral t 0 pffiffi Since wðtÞ :¼ t does not have a continuous derivative on [0, 4], the Substitution Theorem 7.3.8 is not applicable, at least with this substitution. (In fact, it is not obvious that this integral exists; however, we can apply Exercise 7.2.11 to obtainpthis ffiffi conclusion. We could then apply the Fundamental Theorem 7.3.1 to F ðtÞ :¼ 2 cos t with E :¼ f0g to & evaluate this integral.) We will give a more powerful Substitution Theorem for the generalized Riemann integral in Section 10.1. Lebesgue’s lntegrability Criterion We will now present a statement of the definitive theorem due to Henri Lebesgue (1875–1941) giving a necessary and sufficient condition for a function to be Riemann integrable, and will give some applications of this theorem. In order to state this result, we need to introduce the important notion of a null set. Warning Some people use the term ‘‘null set’’ as a synonym for the terms ‘‘empty set’’ or ‘‘void set’’ referring to ; (¼ the set that has no elements). However, we will always use the term ‘‘null set’’ in conformity with our next definition, as is customary in the theory of integration. 7.3.10 Definition (a) A set Z R is said to be a null set if for every e > 0 there exists a countable collection fðak ; bk Þg1 k¼1 of open intervals such that 1 1 [ X Z ð ak ; bk Þ and ðbk ak Þ e: ð6Þ k¼1 k¼1 C07 12/13/2010 10:5:38 Page 221 7.3 THE FUNDAMENTAL THEOREM 221 (b) If Q(x) is a statement about the point x 2 I, we say that Q(x) holds almost everywhere on I (or for almost every x 2 I), if there exists a null set Z I such that Q(x) holds for all x 2 InZ. In this case we may write QðxÞ for a:e: x 2 I: It is trivial that any subset of a null set is also a null set, and it is easy to see that the union of two null sets is a null set. We will now give an example that may be very surprising. 7.3.11 Example The Q 1 of rational numbers in [0, 1] is a null set. We enumerate Q 1 ¼ fr1 ; r2 ; . . .g. Given e > 0, note that the open interval J 1 :¼ ðr1 e=4; r1 þ e=4Þ contains r1 and has length e=2; also the open interval J 2 :¼ ðr2 e=8; r2 þ e=8) contains r2 and has length e=4. In general, the open interval e e J k :¼ rk kþ1 ; rk þ kþ1 2 2 S contains the point rk and has length e=2k . Therefore, the union 1 k¼1 J k of these open 1 X intervals contains every point of Q 1 ; moreover, the sum of the lengths is e=2k ¼ e. & k¼1 Since e > 0 is arbitrary, Q 1 is a null set. The argument just given can be modified to show that: Every countable set is a null set. However, it can be shown that there exist uncountable null sets in R; for example, the Cantor set that will be introduced in Definition 11.1.10. We now state Lebesgue’s Integrability Criterion. It asserts that a bounded function on an interval is Riemann integrable if and only if its points of discontinuity form a null set. 7.3.12 Lebesgue’s Integrability Criterion A bounded function f : ½a; b ! R is Riemann integrable if and only if it is continuous almost everywhere on [a, b]. A proof of this result will be given in Appendix C. However, we will apply Lebesgue’s Theorem here to some specific functions, and show that some of our previous results follow immediately from it. We shall also use this theorem to obtain the important Composition and Product Theorems. 7.3.13 Examples (a) The step function g in Example 7.1.4(b) is continuous at every point except the point x ¼ 1. Therefore it follows from the Lebesgue Integrability Criterion that g is Riemann integrable. In fact, since every step function has at most a finite set of points of discontinuity, then: Every step function on [a, b] is Riemann integrable. (b) Since it was seen in Theorem 5.6.4 that the set of points of discontinuity of a monotone function is countable, we conclude that: Every monotone function on [a, b] is Riemann integrable. (c) The function G in Example 7.1.4(d) is discontinuous precisely at the points D :¼ f1; 1=2; . . . ; 1=n; . . .g. Since this is a countable set, it is a null set and Lebesgue’s Criterion implies that G is Riemann integrable. (d) The Dirichlet function was shown in Example 7.2.2(b) not to be Riemann integrable. Note that it is discontinuous at every point of [0, 1]. Since it can be shown that the interval [0, 1] is not a null set, Lebesgue’s Criterion yields the same conclusion. C07 12/13/2010 10:5:38 222 Page 222 CHAPTER 7 THE RIEMANN INTEGRAL (e) Let h : ½0; 1 ! R be Thomae’s function, defined in Examples 5.1.6(h) and 7.l.7. In Example 5.1.6(h), we saw that h is continuous at every irrational number and is discontinuous at every rational number in [0, 1]. By Example 7.3.11, it is discontinuous on a null set, so Lebesgue’s Criterion implies that Thomae’s function is Riemann integrable on & [0, 1], as we saw in Example 7.1.7. We now obtain a result that will enable us to take other combinations of Riemann integrable functions. 7.3.14 Composition Theorem Let f 2 R½a; b with f ð½a; bÞ ½c; d and let w : ½c; d ! R be continuous. Then the composition w f belongs to R½a; b. Proof. If f is continuous at a point u 2 ½a; b, then w f is also continuous at u. Since the set D of points of discontinuity of f is a null set, it follows that the set D1 D of points of discontinuity of w f is also a null set. Therefore the composition w f also belongs to R½a; b. Q.E.D. It will be seen in Exercise 22 that the hypothesis that w is continuous cannot be dropped. The next result is a corollary of the Composition Theorem. 7.3.15 Corollary Suppose that f 2 R½a; b. Then its absolute value j f j is in R½a; b, and Z b Z b f j f j M ðb aÞ; a a where j f ðxÞj M for all x 2 ½a; b. Proof. We have seen in Theorem 7.1.6 that if f is integrable, then there exists M such that j f ðxÞj M for all x 2 ½a; b. Let wðtÞ :¼ jtj for t 2 ½M; M ; then the Composition Theorem implies that j f j ¼ w f 2 R½a; b. The first inequality follows from the fact that j f j f j f j and 7.1.5(c), and the second from the fact that j f ðxÞj M. Q.E.D. 7.3.16 The Product Theorem If f and g belong to R½a; b, then the product fg belongs to R½a; b. Proof. If wðtÞ :¼ t2 for t 2 ½M; M , it follows from the Composition Theorem that f 2 ¼ w f belongs to R½a; b. Similarly, ð f þ gÞ2 and g2 belong to R½a; b. But since we can write the product as i 1h f g ¼ ð f þ gÞ 2 f 2 g2 ; 2 it follows that f g 2 R½a; b. Q.E.D. Integration by Parts We will conclude this section with a rather general form of Integration by Parts for the Riemann integral, and Taylor’s Theorem with the Remainder. 7.3.17 Integration by Parts Let F,G be differentiable on [a, b] and let f :¼ F 0 and g :¼ G0 belong to R½a; b . Then Z b Z b b ð7Þ f G ¼ FG Fg: a a a C07 12/13/2010 10:5:38 Page 223 7.3 THE FUNDAMENTAL THEOREM 223 Proof. By Theorem 6.1.3(c), the derivative ðFGÞ0 exists on [a, b] and ðFGÞ0 ¼ F 0 G þ FG0 ¼ f G þ Fg: Since F, G are continuous and f, g belong to R½a; b, the Product Theorem 7.3.16 implies that fG and Fg are integrable. Therefore the Fundamental Theorem 7.3.1 implies that Z b Z b Z b b FG ¼ ðFGÞ0 ¼ fG þ Fg; a a a a from which (7) follows. Q.E.D. A special, but useful, case of this theorem R x is when f and g are R x continuous on [a, b] and F, G are their indefinite integrals F ðxÞ :¼ a f and GðxÞ :¼ a g. We close this section with a version of Taylor’s Theorem for the Riemann Integral. 7.3.18 Taylor’s Theorem with the Remainder Suppose that f 0 ; . . . ; f ðnÞ ; f ðnþ1Þ exist on [a, b] and that f ðnþ1Þ 2 R½a; b. Then we have ð8Þ f ð bÞ ¼ f ð aÞ þ f 0 ð aÞ f ðnÞ ðaÞ ð b aÞ þ þ ðb aÞn þ Rn ; 1! n! where the remainder is given by ð9Þ Rn ¼ 1 n! Z a b f ðnþ1Þ ðtÞ ðb tÞn dt: Proof. Apply Integration by Parts to equation (9), with F ðtÞ :¼ f ðnÞ ðtÞ and GðtÞ :¼ ðb tÞn =n!, so that gðtÞ ¼ ðb tÞn1 =ðn 1Þ!, to get Z b t¼b 1 1 Rn ¼ f ðnÞ ðtÞ ðb tÞn þ f ðnÞ ðtÞ ðb aÞn1 dt t¼a n! ðn 1Þ! a Z b f ðnÞ ðaÞ 1 n ð b aÞ þ ¼ f ðnÞ ðtÞ ðb tÞn1 dt: n! ðn 1Þ! a If we continue to integrate by parts in this way, we obtain (8). Q.E.D. Exercises for Section 7.3 1. Extend the proof of the Fundamental Theorem 7.3.1 to the case of an arbitrary finite set E. 2. If n 2 N andRH n ðxÞ :¼ xnþ1 =ðn þ 1Þ for x 2 ½a; b, show that the Fundamental Theorem 7.3.1 b implies that a xn dx ¼ bnþ1 anþ1 =ðn þ 1Þ. What is the set E here? 1 2 3. If R 3 gðxÞ :¼ x for jxj 1 and gðxÞ :¼ x for jxj < 1 and if GðxÞ :¼ 2 x 1 , show that g ð x Þdx ¼ G ð 3 Þ G ð 2 Þ ¼ 5=2. 2 Rb 4. Let BðxÞ :¼ 12 x2 for x < 0 and BðxÞ :¼ 12 x2 for x 0. Show that a jxjdx ¼ BðbÞ BðaÞ. 5. Let f : ½a; b ! R and let C 2 R. (a) If F : ½a; b ! R is an antiderivative of f on [a, b], show that FC ðxÞ :¼ FðxÞ þ C is also an antiderivative of f on [a, b]. (b) If F1 and F2 are antiderivatives of f on [a, b], show that F1 F2 is a constant function on [a, b]. C07 12/13/2010 10:5:39 224 Page 224 CHAPTER 7 THE RIEMANN INTEGRAL Rz 6. If f 2 R½a; b and if c 2 ½a; b, the function defined by F c ðzÞ :¼ c f for z 2 ½a; b is called the indefinite integral of f with basepoint c. Find a relation between Fa and Fc. 7. We have seen in Example 7.1.7 that Thomae’s function is in R½0; 1 with integral equal to 0. Can the Fundamental Theorem 7.3.1 be used to obtain this conclusion? Explain your answer. 8. Let F(x) be defined for x 0 by F ðxÞ :¼ ðn 1Þx ðn 1Þn=2 for x 2 ½n 1; nÞ; n 2 N. Show that F is continuous and evaluate F 0 ðxÞ at points where this derivative exists. Use this Rb result to evaluate a ½½xdx for 0 a < b, where ½½x denotes the greatest integer in x, as defined in Exercise 5.1.4. Rx 9. Let f 2 R½a; b and define R x F ðxÞ :¼ a f for x 2 ½a; b. (a) Evaluate GðxÞ :¼ c f in terms of F, where c 2 ½a; b. Rb (b) Evaluate H ðxÞ :¼ x f in terms of F. R sin x (c) Evaluate SðxÞ :¼ x f in terms of F. 10. Let f : ½a; b ! R be continuous on [a, b] and v : ½c; d ! R be differentiable on [c, d] with R vðxlet Þ vð½c; d Þ ½a; b. If we define GðxÞ :¼ a f , show that G0 ðxÞ ¼ f ðvðxÞÞ v0 ðxÞ for all x 2 ½c; d . on [0, 1] by: 11. Find F 0 ðxÞ when R x2 F is defined 1 (a) F ðxÞ :¼ 0 ð1 þ t3 Þ dt: (b) F ðxÞ :¼ R x pffiffiffiffiffiffiffiffiffiffiffiffi 1 þ t2 dt: x2 12. Let f : ½0; 3 ! R be defined by f ðxÞ :¼ x for 0 x < 1; f ðxÞ :¼ 1 for 1 x < 2 and f ðxÞ :¼ x Rx for 2 x 3. Obtain formulas for F ðxÞ :¼ 0 f and sketch the graphs of f and F. Where is F differentiable? Evaluate F 0 ðxÞ at all such points. 13. The function g is defined on [0, 3] by gðxÞ :¼ 1 if 0 x < 2 and gðxÞ :¼ 1 if 2 x 3. Find Rx the indefinite integral GðxÞ ¼ 0 g for 0 x 3, and sketch the graphs of g and G. Does G0 ðxÞ ¼ gðxÞ for all x in [0, 3]? 14. Show there does not exist a continuously differentiable function f on [0, 2] such that f ð0Þ ¼ 1; f ð2Þ ¼ 4, and f 0 ðxÞ 2 for 0 x 2. (Apply the Fundamental Theorem.) R xþc 15. If f : R ! R is continuous and c > 0, define g : R ! R by gðxÞ :¼ xc f ðtÞdt. Show that g is 0 differentiable on R and find g ðxÞ. Rx R1 16. If f : ½0; 1 ! R is continuous and 0 f ¼ x f for all x 2 ½0; 1, show that f ðxÞ ¼ 0 for all x 2 ½0; 1. Ru 17. Use the following argument to prove the Substitution Theorem 7.3.8. Define F ðuÞ :¼ wðaÞ f ðxÞdx for u 2 I, and H ðtÞ :¼ F ðwðtÞÞ for t 2 J. Show that H 0 ðtÞ ¼ f ðwðtÞÞw0 ðtÞ for t 2 J and that Z wðbÞ wðaÞ Z f ðxÞdx ¼ F ðwðbÞÞ ¼ H ðbÞ ¼ b f ðwðtÞÞw0 ðtÞdt: a 18. Use the Substitution Theorem 7.3.8 to evaluate the following integrals. Z 2 Z 1 pffiffiffiffiffiffiffiffiffiffiffiffi 1=2 t 1 þ t2 dt; t2 1 þ t3 dt ¼ 4=3; (b) (a) 0 0 pffiffi pffiffi Z 4 Z 4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þ t cos t pffiffi pffiffi dt ¼ 2ðsin 2 sin1Þ: dt; (d) (c) t t 1 1 19. Explain why Theorem 7.3.8 and/or Exercise 7.3.17 cannot be applied to evaluate the following integrals, using the indicated substitution. pffiffi Z 4 pffiffi Z 4 pffiffi pffiffi tdt cos t dt (a) (b) pffiffi wðtÞ ¼ t; pffiffi wðtÞ ¼ t; t t 0 1þ 0 Z 1 Z 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dt (d) (c) pffiffiffiffiffiffiffiffiffiffiffiffi wðtÞ ¼ Arcsin t: 1 þ 2jtjdt wðtÞ ¼ jtj; 1 t2 0 1 C07 12/13/2010 10:5:39 Page 225 7.4 THE DARBOUX INTEGRAL 225 20. (a) If Z1 and Z2 are null sets, show that Z 1 [ Z 2 is a null set. S1 (b) More generally, if Zn is a null n set for each n 2 N, show that n¼1 Z n is a null set. [Hint: Given e > 0 and n 2 N, let J k : k 2 N be a countable collection of open intervals whose n union contains n Zn and the sum of whose lengths is e=2 . Now consider the countable collection J k : n; k 2 N .] 21. Let f ; g 2 R½a; b. Rb (a) If t 2 R, show that a ðtf gÞ2 0. R Rb Rb b (b) Use (a) to show that 2 a f g t a f 2 þ ð1=tÞ a g2 for t > 0. Rb 2 Rb (c) If a f ¼ 0, show that a f g ¼ 0. R 2 Rb Rb Rb b 2 (d) Now prove that a f g a j f gj a f 2 a g2 . This inequality is called the Cauchy-Bunyakovsky-Schwarz Inequality (or simply the Schwarz Inequality). 22. Let h : ½0; 1 ! R be Thomae’s function and let sgn be the signum function. Show that the composite function sgn h is not Riemann integrable on [0, 1]. Section 7.4 The Darboux Integral An alternative approach to the integral is due to the French mathematician Gaston Darboux (1842–1917). Darboux had translated Riemann’s work on integration into French for publication in a French journal and inspired by a remark of Riemann, he developed a treatment of the integral in terms of upper and lower integrals that was published in 1875. Approximating sums in this approach are obtained from partitions using the infima and suprema of function values on subintervals, which need not be attained as function values and thus the sums need not be Riemann sums. This approach is technically simpler in the sense that it avoids the complications of working with infinitely many possible choices of tags. But working with infima and suprema also has its complications, such as lack of additivity of these quantities. Moreover, the reliance on the order properties of the real numbers causes difficulties in extending the Darboux integral to higher dimensions, and, more importantly, impedes generalization to more abstract surfaces such as manifolds. Also, the powerful Henstock-Kurzweil approach to integration presented in Chapter 10, which includes the Lebesgue integral, is based on the Riemann definition as given in Section 7.1. In this section we introduce the upper and lower integrals of a bounded function on an interval, and define a function to be Darboux integrable if these two quantities are equal. We then look at examples and establish a Cauchy-like integrability criterion for the Darboux integral. We conclude the section by proving that the Riemann and Darboux approaches to the integral are in fact equivalent, that is, a function on a closed, bounded interval is Riemann integrable if and only if it is Darboux integrable. Later topics in the book do not depend on the Darboux definition of integral so that this section can be regarded as optional. Upper and Lower Sums Let f : I ! R be a bounded function on I ¼ ½a; b and let P ¼ ðx0 ; x1 ; . . . ; xn Þ be a partition of I. For k ¼ 1; 2; . . . ; n we let mk :¼ inf f f ðxÞ : x 2 ½xk1 ; xk g; M k :¼ supf f ðxÞ : x 2 ½xk1 ; xk g: C07 12/13/2010 10:5:40 226 Page 226 CHAPTER 7 THE RIEMANN INTEGRAL Figure 7.4.1 Lð f ; P Þ a lower sum Figure 7.4.2 U ð f ; P Þ an upper sum The lower sum of f corresponding to the partition P is defined to be Lð f ; P Þ :¼ n X mk ðxk xk1 Þ; k¼1 and the upper sum of f corresponding to P is defined to be U ð f ; P Þ :¼ n X M k ðxk xk1 Þ: k¼1 If f is a positive function, then the lower sum Lð f ; P Þ can be interpreted as the area of the union of rectangles with base ½xk1 ; xk and height mk . (See Figure 7.4.1.) Similarly, the upper sum U ð f ; P Þ can be interpreted as the area of the union of rectangles with base ½xk1 ; xk and height M k . (See Figure 7.4.2.) The geometric interpretation suggests that, for a given partition, the lower sum is less than or equal to the upper sum. We now show this to be the case. 7.4.1 Lemma If f :¼ I ! R is bounded and P is any partition of I, then Lð f ; P Þ U ð f ; P Þ. Proof. Let P :¼ ðx0 ; x1 ; . . . ; xn Þ. Since mk M k for k ¼ 1; 2; . . . ; n and since xk xk1 > 0 for k ¼ 1; 2; . . . ; n; it follows that Lð f ; P Þ ¼ n X mk ðxk xk1 Þ k¼1 n X M k ðxk xk1 Þ ¼ U ð f ; P Þ: Q.E.D. k¼1 If P :¼ ðx0 ; x1 ; . . . ; xn Þ and Q :¼ ðy0 ; y1 ; . . . ; ym Þ are partitions of I, we say that Q is a refinement of P if each partition point xk 2 P also belongs to Q (that is, if P Q). A refinement Q of a partition P can be obtained by adjoining a finite number of points to P. In this case, each one of the intervals ½xk1 ; xk into which P divides I can be written as the union of intervals whose end points belong to Q; that is, ½xk1 ; xk ¼ yj1 ; yj [ yj ; yjþ1 [ [ ½yh1 ; yh : We now show that refining a partition increases lower sums and decreases upper sums. 7.4.2 Lemma If f : I ! R is bounded, if P is a partition of I, and if Q is a refinement of P, then Lð f ; P Þ Lð f ; QÞ and U ð f ; QÞ U ð f ; P Þ Proof. Let P ¼ ðx0 ; x1 ; . . . ; xn Þ. We first examine the effect of adjoining one point to P. Let z 2 I satisfy xk1 < z < xk and let P 0 be the partition P 0 :¼ ðx0 ; x1 ; . . . ; xk1 ; z; xk ; . . . ; xn Þ; C07 12/13/2010 10:5:40 Page 227 7.4 THE DARBOUX INTEGRAL 227 obtained from P by adjoining z to P. Let m0k and m00k be the numbers m0k :¼ inf f f ðxÞ : x 2 ½xk1 ; zg; m00k :¼ inf f f ðxÞ : x 2 ½z; xk g: Then mk m0k and mk m00k (why?) and therefore mk ðxk xk1 Þ ¼ mk ðz xk1 Þ þ mk ðxk zÞ m0k ðz xk1 Þ þ m00k ðxk zÞ: If we add the terms mj xj xj1 for j 6¼ k to the above inequality, we obtain Lð f ; P Þ L ð f ; P 0 Þ. Now if Q is any refinement of P (i.e., if P Q), then Q can be obtained from P by adjoining a finite number of points to P one at a time. Hence, repeating the preceding argument, we infer that Lð f ; P Þ Lð f ; QÞ. Upper sums are handled similarly; we leave the details as an exercise. Q.E.D. These two results are now combined to conclude that a lower sum is always smaller than an upper sum even if they correspond to different partitions. 7.4.3 Lemma Let f : I ! R be bounded. If P 1 ; P 2 are any two partitions of I, then Lð f ; P 1 Þ U ð f ; P 2 Þ. Proof. Let Q :¼ P 1 [ P 2 be the partition obtained by combining the points of P 1 and P 2 . Then Q is a refinement of both P 1 and P 2 . Hence, by Lemmas 7.4.1 and 7.4.2, we conclude that Lð f ; P 1 Þ Lð f ; QÞ U ð f ; QÞ U ð f ; P 2 Þ: Q.E.D. Upper and Lower Integrals We shall denote the collection of all partitions of the interval I by PðI Þ. If f : I ! R is bounded, then each P in PðI Þ determines two numbers: Lð f ; P Þ and U ð f ; P Þ. Thus, the collection PðI Þ determines two sets of numbers: the set of lower sums Lð f ; P Þ for P 2 PðI Þ, and the set of upper sums U ð f ; P Þ for P 2 PðI Þ. Hence, we are led to the following definitions. 7.4.4 Definition Let I :¼ ½a; b and let f : I ! R be a bounded function. The lower integral of f on I is the number Lð f Þ :¼ supfLð f ; P Þ : P 2 PðI Þg; and the upper integral of f on I is the number U ð f Þ :¼ inf fU ð f ; P Þ : P 2 PðI Þg: Since f is a bounded function, we are assured of the existence of the numbers mI :¼ inf f f ðxÞ : x 2 I g and M I :¼ supf f ðxÞ : x 2 I g: It is readily seen that for any P 2 PðI Þ, we have mI ðb aÞ Lð f ; P Þ U ð f ; P Þ M I ðb aÞ: Hence it follows that ð1Þ m I ð b aÞ L ð f Þ The next inequality is also anticipated. and U ð f Þ M I ðb aÞ: C07 12/13/2010 10:5:40 228 Page 228 CHAPTER 7 THE RIEMANN INTEGRAL 7.4.5 Theorem Let I ¼ ½a; b and let f : I ! R be a bounded function. Then the lower integral Lð f Þ and the upper integral U ð f Þ of f on I exist. Moreover, ð2Þ Lð f Þ U ð f Þ: Proof. If P 1 and P 2 are any partitions of I, then it follows from Lemma 7.4.3 that Lð f ; P 1 Þ U ð f ; P 2 Þ. Therefore the number U ð f ; P 2 Þ is an upper bound for the set fLð f ; P Þ : P 2 PðI Þg. Consequently, Lð f Þ, being the supremum of this set, satisfies Lð f Þ U ð f ; P 2 Þ. Since P 2 is an arbitrary partition of I, then Lð f Þ is a lower bound for the set fU ð f ; P Þ : P 2 PðI Þg. Consequently, the infimum U ð f Þ of this set satisfies the inequality (2). Q.E.D. The Darboux Integral If I is a closed bounded interval and f : I ! R is a bounded function, we have proved in Theorem 7.4.5 that the lower integral Lð f Þ and the upper integral U ð f Þ always exist. Moreover, we always have Lð f Þ U ð f Þ. However, it is possible that we might have Lð f Þ < U ð f Þ, as we will see in Example 7.4.7(d). On the other hand, there is a large class of functions for which Lð f Þ ¼ U ð f Þ. 7.4.6 Definition Let I ¼ ½a; b and let f : I ! R be a bounded function. Then f is said to be Darboux integrable on I if Lð f Þ ¼ U ð f Þ. In this case the Darboux integral of f over I is defined to be the value Lð f Þ ¼ U ð f Þ. Thus we see that if the Darboux integral of a function on an interval exists, then the integral is the unique real number that lies between the lower sums and the upper sums. Since we will soon establishRthe equivalence of the Darboux and Riemann integrals, Rb b we will use the standard notation a f or a f ðxÞ dx for the Darboux integral of a function f on ½a; b. The context should prevent any confusion from arising. 7.4.7 Examples (a) A constant function is Darboux integrable. Let f ðxÞ :¼ c for x 2 I :¼ ½a; b. If P is any partition of I, it is easy to see that Lð f ; P Þ ¼ cðb aÞ ¼ U ð f ; P Þ (See Exercise 7.4.2). Therefore the lower and upper integrals Rb R bare given by Lð f Þ ¼ cðb aÞ ¼ U ð f Þ. Consequently, f is integrable on I and f ¼ a a c dx ¼ cðb aÞ: (b) Let g be defined on [0, 3] as follows: gðxÞ :¼ 2 if 0 x 1 and gðxÞ :¼ 3 if 2 < x 3. (See Example 7.1.4(b).) For e > 0, if we define the partition P e :¼ ð0; 1; 1 þ e; 3Þ, then we get the upper sum U ðg; P e Þ ¼ 2 ð1 0Þ þ 3ð1 þ e 1Þ þ 3ð2 eÞ ¼ 2 þ 3e þ 6 3e ¼ 8: Therefore, the upper integral satisfies U ðgÞ 8. (Note that we cannot yet claim equality because U ðgÞ is the infimum over all partitions of [0, 3].) Similarly, we get the lower sum Lðg; P e Þ ¼ 2 þ 2e þ 3ð2 eÞ ¼ 8 e; so that the lower integral satisfies LðgÞ 8. Then we haveR8 LðgÞ U ðgÞ 8, and 3 hence LðgÞ ¼ U ðgÞ ¼ 8. Thus the Darboux integral of g is 0 g ¼ 8. C07 12/13/2010 10:5:41 Page 229 7.4 THE DARBOUX INTEGRAL 229 (c) The function hðxÞ :¼ x is integrable on [0, 1]. Let P n be the partition of I :¼ ½0; 1 into n subintervals given by 1 2 n1 n ; ¼1 : P n :¼ 0; ; ; . . . ; n n n n Since h is an increasing function, its infimum and supremum on the subinterval ½ðk 1Þ=n; k=n are attained at the left and right end points, respectively, and are thus given by mk ¼ ðk 1Þ=n and M k ¼ k=n. Moreover, since xk xk1 ¼ 1=n for all k ¼ 1; 2; . . . ; n, we have Lðh; P n Þ ¼ ð0 þ 1 þ þ ðn 1ÞÞ=n2 ; U ðh; P n Þ ¼ ð1 þ 2 þ þ nÞ=n2 : If we use the formula 1 þ 2 þ þ m ¼ mðm þ 1Þ=2, for m 2 N, we obtain ðn 1Þn 1 1 nð n þ 1Þ 1 1 1 ; U ðh; P n Þ ¼ 1þ : ¼ ¼ Lðh; P n Þ ¼ 2n2 2 n 2n2 2 n Since the set of partitions fP n : n 2 N g is a subset of the set of all partitions of P ðI Þ of I, it follows that 1 ¼ supfLðh; P n Þ : n 2 N g supfLðh; P Þ : P 2 PðI Þg ¼ LðhÞ; 2 and also that 1 U ðhÞ ¼ inf fU ðh; P Þ : P 2 PðI Þg inf fU ðh; P n Þ : n 2 N g ¼ : 2 Since 12 LðhÞ U ðhÞ 12, we conclude that LðhÞ ¼ U ðhÞ ¼ 12. Therefore h is R1 R1 Darboux integrable on I ¼ [0, 1] and 0 h ¼ 0 x dx ¼ 12 : (d) A nonintegrable function. Let I :¼ ½0; 1 and let f : I ! R be the Dirichlet function defined by f ðxÞ :¼ 1 :¼ 0 for for x rational; x irrational: If P :¼ ðx0 ; x1 ; . . . ; xn Þ is any partition of [0, 1], then since every nontrivial interval contains both rational numbers and irrational numbers (see the Density Theorem 2.4.8 and its corollary), we have mk ¼ 0 and M k ¼ 1. Therefore, we have Lð f ; P Þ ¼ 0; U ð f ; P Þ ¼ 1; for all P 2 PðI Þ, so that Lð f Þ ¼ 0; U ð f Þ ¼ 1: Since Lð f Þ 6¼ U ð f Þ, the function f is not Darboux integrable on [0, 1]. We now establish some conditions for the existence of the integral. 7.4.8 Integrability Criterion Let I :¼ ½a; b and let f : I ! R be a bounded function on I. Then f is Darboux integrable on I if and only if for each e > 0 there is a partition P e of I such that ð3Þ U ð f ; P e Þ Lð f ; P e Þ < e: Proof. If f is integrable, then we have Lð f Þ ¼ U ð f Þ. If e > 0 is given, then from the definition of the lower integral as a supremum, there is a partition P 1 of I such that Lð f Þ e=2 < Lð f ; P 1 Þ: Similarly, there is a partition P 2 of I such that U ð f ; P 2 Þ < U ð f Þ þ e=2: If we let P e :¼ P 1 [ P 2 , then P e is a refinement of both P 1 and P 2 . Consequently, by C07 12/13/2010 10:5:41 230 Page 230 CHAPTER 7 THE RIEMANN INTEGRAL Lemmas 7.4.1 and 7.4.2, we have Lð f Þ e=2 < Lð f ; P 1 Þ Lð f ; Pe Þ U ð f ; P e Þ U ð f ; P 2 Þ < U ð f Þ þ e=2: Since Lð f Þ ¼ U ð f Þ, we conclude that (3) holds. To establish the converse, we first observe that for any partition P we have Lð f ; P Þ Lð f Þ and U ð f Þ U ð f ; P Þ. Therefore, U ð f Þ Lð f Þ U ð f ; P Þ Lð f ; P Þ: Now suppose that for each e > 0 there exists a partition P e such that (3) holds. Then we have U ð f Þ Lð f Þ U ð f ; P e Þ Lð f ; P e Þ < e: Since e > 0 is arbitrary, we conclude that U ð f Þ Lð f Þ. Since the inequality Lð f Þ U ð f Þ is always valid, we have Lð f Þ ¼ U ð f Þ. Hence f is Darboux integrable. Q.E.D. 7.4.9 Corollary Let I ¼ ½a; b and let f : I ! R be a bounded function. If fPn : n 2 N g is a sequence of partitions of I such that lim ðU ð f ; Pn Þ Lð f ; Pn ÞÞ ¼ 0; n Rb then f is integrable and lim Lð f ; P n Þ ¼ a f ¼ limn U ð f ; P n Þ. n Proof. If e > 0 is given, it follows from the hypothesis that there exists K such that if n K then U ð f ; P n Þ Lð f ; P n Þ < e, whence the integrability of f follows from the Integrability Criterion. We leave the remainder of the proof as an exercise. Q.E.D. The significance of the corollary is the fact that although the definition of the Darboux integral involves the set of all possible partitions of an interval, for a given function, the existence of the integral and its value can often be determined by a special sequence of partitions. For example, if hðxÞ ¼ x on [0, 1] and P n is the partition as in Example 7.4.7(c), then and therefore R1 0 limðU ðh; P n Þ Lðh; P n ÞÞ ¼ lim 1=n ¼ 0 x dx ¼ lim U ðh; P n Þ ¼ lim 12 ð1 þ 1=nÞ ¼ 12 : Continuous and Monotone Functions It was shown in Section 7.2 that functions that are continuous or monotone on a closed bounded interval are Riemann integrable. (See Theorems 7.2.7 and 7.2.8.) The proofs employed approximation by step functions and the Squeeze Theorem 7.2.3 as the main tools. Both proofs made essential use of the fact that both continuous functions and monotone functions attain a maximum value and a minimum value on a closed bounded interval. That is, if f is a continuous or monotone function on [a, b], then for a partition P ¼ ðx0 ; x1 ; . . . ; xn Þ, the numbers M k ¼ supf f ðxÞ : x 2 I k g and mk ¼ inf f f ðxÞ : x 2 I k g, k ¼ 1; 2; . . . ; n, are attained as function values. For continuous functions, this is Theorem 5.3.4, and for monotone functions, these values are attained at the right and left endpoints of the interval. If we define the step function v on [a, b] by vðxÞ :¼ M k for x 2 ½xk1 ; xk Þ for k ¼ 1; 2; . . . ; n 1, and vðxÞ :¼ M n for x 2 ½xn1 ; xn , then we observe that the Riemann C07 12/13/2010 10:5:42 Page 231 7.4 THE DARBOUX INTEGRAL integral of v is given by Rb a v¼ n P k¼1 231 M k ðxk xk1 Þ. (See Theorem 7.2.5.) Now we recognize the sum on the right as the upper Darboux sum U ð f ; P Þ, so that we have Z b n X v¼ M k ðxk xk1 Þ ¼ U ð f ; P Þ: a k¼1 Similarly, if the step function a is defined by aðxÞ :¼ mk for x 2 ½xk1 ; xk Þ, k ¼ 1; 2; . . . ; n 1, and aðxÞ :¼ mn for x 2 ½xn1 ; xn , then we have the Riemann integral Z b n X a¼ mk ðxk xk1 Þ ¼ Lð f ; P Þ: a k¼1 Subtraction then gives us Z b n X ðv a Þ ¼ ðM k mk Þðxk xk1 Þ ¼ U ð f ; P Þ Lð f ; P Þ: a k¼1 We thus see that the Integrability Criterion 7.4.8 is the Darboux integral counterpart to the Squeeze Theorem 7.2.3 for the Riemann integral. Therefore, if we examine the proofs of Theorems 7.2.7 and 7.2.8 that establish the Riemann integrability of continuous and monotone functions, respectively, and replace the integrals of step functions by the corresponding lower and upper sums, then we obtain proofs of the theorems for the Darboux integral. (For example, in Theorem 7.2.7 for continuous functions, we would have ae ðxÞ ¼ f ðui Þ ¼ mi and ve ðxÞ ¼ f ðvi Þ ¼ M i and replace the integral of ve ae with U ð f ; P Þ Lð f ; P Þ:) Thus we have the following theorem. We leave it as an exercise for the reader to write out the proof. 7.4.10 Theorem If the function f on the interval I ¼ [a, b] is either continuous or monotone on I, then f is Darboux integrable on I. The preceding observation that connects the Riemann and Darboux integrals plays a role in the proof of the equivalence of the two approaches to integration, which we now discuss. Of course, once equivalence has been established, then the preceding theorem would be an immediate consequence. Equivalence We conclude this section with a proof that the Riemann and Darboux definitions of the integral are equivalent in the sense that a function on a closed, bounded interval is Riemann integrable if and only if it is Darboux integrable, and their integrals are equal. This is not immediately apparent. The Riemann integral is defined in terms of sums that use function values (tags) together with a limiting process based on the length of subintervals in a partition. On the other hand, the Darboux integral is defined in terms of sums that use infima and suprema of function values, which need not be function values, and a limiting process based on refinement of partitions, not the size of subintervals in a partition. Yet the two are equivalent. The background needed to prove equivalence is at hand. For example, if a function is Darboux integrable, we recognize that upper and lower Darboux sums are Riemann integrals of step functions. Thus the Integrability Criterion 7.4.8 for the Darboux integral C07 12/13/2010 10:5:42 232 Page 232 CHAPTER 7 THE RIEMANN INTEGRAL corresponds to the Squeeze Theorem 7.2.3 for the Riemann integral in its application. In the other direction, if a function is Riemann integrable, the definitions of supremum and infimum enable us to choose tags to obtain Riemann sums that are as close to upper and lower Darboux sums as we wish. In this way, we connect the Riemann integral to the upper and lower Darboux integrals. The details are given in the proof. 7.4.11 Equivalence Theorem A function f on I ¼ [a, b] is Darboux integrable if and only if it is Riemann integrable. Proof. Assume that f is Darboux integrable. For e > 0, let P e be a partition of [a, b] such that U ð f ; P e Þ Lð f ; P e Þ < e. For this partition, as in the preceding discussion, we define the step functions ae and ve on [a, b] by ae ðxÞ :¼ mk and ve ðxÞ :¼ M k for x 2 ½xk1 ; xk Þ; k ¼ 1; 2; . . . ; n 1, and ae ðxÞ :¼ mn ; ve ðxÞ :¼ M n for x 2 ½xn1 ; xn , where, as usual, M k is the supremum and mk the infimum of f on I k ¼ ½xk1 ; xk . Clearly we have ð4Þ ae ðxÞ f ðxÞ ve ðxÞ for all x in ½a; b: Moreover, by Theorem 7.2.5, these functions are Riemann integrable and their integrals are equal to Z b Z b n n P P ve ¼ M k ðxk xk1 Þ ¼ U ð f ; P e Þ; ae ¼ mk ðxk xk1 Þ ¼ Lð f ; P e Þ: ð5Þ a a k¼1 Therefore, we have Z a b k¼1 ðve ae Þ ¼ U ð f ; P e Þ Lð f ; P e Þ < e: By the Squeeze Theorem 7.2.3, it follows that f is Riemann integrable. Moreover, we note that (4) and (5) are valid for any partition P and therefore the Riemann integral of f lies between Lð f ; P Þ and U ð f ; P Þ for any partition P. Therefore the Riemann integral of f is equal to the Darboux integral of f. Rb Now assume that f is Riemann integrable and let A ¼ a f denote the value of the integral. Then, f is bounded by Theorem 7.1.6, and given e> 0, there exists d > 0 such that _ < d, we have S f ; P_ A < e, which can be written for any tagged partition P_ with jjPjj ð6Þ A e < S f ; P_ < A þ e: If P ¼ ðx0 ; x1 ; . . . ; xn Þ, then because M k ¼ supff ðxÞ : x 2 I k g is a supremum, we can choose tags tk in I k such that f ðtk Þ > M k e=ðb aÞ. Summing, and noting that n P ðxk xk1 Þ ¼ b a; we obtain k¼1 n n P P f ðtk Þðxk xk1 Þ > M k ðxk xk1 Þe ¼ U ð f ; P Þe U ð f Þe: ð7Þ S f ; P_ ¼ k¼1 k¼1 Combining inequalities (6) and (7), we get A þ e > S f ; P_ U ð f Þ e; and hence we have U ð f Þ < A þ 2e. Since e > 0 is arbitrary, this implies that U ð f Þ A. In the same manner, we can approximate lower sums by Riemann sums and show that Lð f Þ > A 2e for arbitrary e > 0, which implies Lð f Þ A. Thus we have R b obtained the inequality A Lð f Þ U ð f Þ A, which gives us Lð f Þ ¼ U ð f Þ ¼ A ¼ a f . Hence, the function f is Darboux integrable with value equal to the Riemann integral. Q.E.D. C07 12/13/2010 10:5:43 Page 233 7.5 APPROXIMATE INTEGRATION 233 Exercises for Section 7.4 1. Let f ðxÞ :¼ jxj for 1 x 2. Calculate Lð f ; P Þ and U ð f ; P Þ for the following partitions: (a) P 1 :¼ ð1; 0; 1; 2Þ, (b) P 2 :¼ ð1; 1=2; 0; 1=2; 1; 3=2; 2Þ. 2. Prove if f ðxÞ :¼ c for x 2 ½a; b, then its Darboux integral is equal to cðb aÞ. 3. Let f and g be bounded functions on I :¼ ½a; b. If f ðxÞ gðxÞ for all x 2 I, show that Lð f Þ LðgÞ and U ð f Þ U ðgÞ. 4. Let f be bounded on [a, b] and let k > 0. Show that Lðkf Þ ¼ kLð f Þ and U ðkf Þ ¼ kU ð f Þ. 5. Let f, g, h be bounded functions on I :¼ ½a; bR such that R b f ðxÞ gðxÞ hðxÞ for all x 2 I. Show b that if f and h are Darboux integrable and if f ¼ a a h, then g is also Darboux integrable with Rb Rb g ¼ f . a a 6. Let f be defined on [0, 2] by f ðxÞ :¼ 1 if x 6¼ 1 and f ð1Þ :¼ 0. Show that the Darboux integral exists and find its value. 7. (a) Prove that if gðxÞ :¼ 0 for 0 x 12 and gðxÞ :¼ 1 for 12 < x 1, then the Darboux integral of g on [0, 1] is equal to 12. (b) Does the conclusion hold if we change the value of g at the point 12 to 13? 8. Let f be continuous on I :¼ ½a; b and assume f ðxÞ 0 for all x 2 I. Prove if Lð f Þ ¼ 0, then f ðxÞ ¼ 0 for all x 2 I. 9. Let f 1 and f 2 be bounded functions on ½a; b. Show that Lð f 1 Þ þ Lð f 2 Þ Lð f 1 þ f 2 Þ. 10. Give an example to show that strict inequality can hold in the preceding exercise. 11. If f is a bounded function on [a, b] such that f ðxÞ ¼ 0 except for x in fc1 ; c2 ; . . . ; cn g in [a, b], show that U ð f Þ ¼ Lð f Þ ¼ 0. 12. Let f ðxÞ ¼ x2 for 0 x 1. For the partition P n :¼ ð0; 1=n; 2=n; . . . ; ðn 1Þ=n; 1Þ, calculate Lð f ; P n Þ and U ð f ; P n Þ, and show that Lð f Þ ¼ U ð f Þ ¼ 13. ðUse the formula 12 þ 22 þ þ m2 ¼ 16 mðm þ 1Þð2m þ 1Þ:Þ 13. Let P e be the partition whose existence is asserted in the Integrability Criterion 7.4.8. Show that if P is any refinement of P e , then U ð f ; P Þ Lð f ; P Þ < e. 14. Write out the proofs that a function f on [a, b] is Darboux integrable if it is either (a) continuous, or (b) monotone. 15. Let f be defined on I :¼ ½a; b and assume that f satisfies the Lipschitz condition j f ðxÞ f ðyÞj K jx yj for all x; y in I. If P n is the partition of I into n equal parts, show Rb that 0 U ð f ; P n Þ a f K ðb aÞ2 =n: Section 7.5 Approximate Integration The Fundamental Theorem of Calculus 7.3.1 yields an effective method of evaluating the Rb integral a f provided we can find an antiderivative F such that F 0 ðxÞ ¼ f ðxÞ when x 2 ½a; b. However, when we cannot find such an F, we may not be able to use the Fundamental Theorem. Nevertheless, when R b f is continuous, there are a number of techniques for approximating the Riemann integral a f by using sums that resemble R b the Riemann sums. One very elementary procedure to obtain quick estimates of a f , based on Theorem 7.1.5(c), is to note that if gðxÞ f ðxÞ hðxÞ for all x 2 ½a; b, then Z b Z b Z b g f h: a a a C07 12/13/2010 10:5:43 234 Page 234 CHAPTER 7 THE RIEMANN INTEGRAL Rb If the integrals of g and h can be calculated, then we have bounds for a f. Often these bounds are accurate enough for our needs. R1 2 For example, suppose we wish to estimate the value of 0 ex dx. It is easy to show 2 that ex ex 1 for x 2 ½0; 1, so that Z 1 Z 1 Z 1 2 ex dx ex dx 1 dx: 0 R1 0 0 Consequently, we have 1 1=e 0 ex dx 1. If we use the mean of the bracketing values, we obtain the estimate 1 1=2e 0:816 for the integral with an error less than 1=2e < 0:184. This estimate is crude, but it is obtained rapidly and may be quite satisfactory for our needs. If a better approximation is desired, we can attempt to find closer approximating functions g and h. 2 Taylor’s Theorem 6.4.1 can be used to approximate ex by a polynomial. In using Taylor’s Theorem, we must get bounds on the remainder term for our calculations to have significance. For example, if we apply Taylor’s Theorem to ey for 0 y 1, we get 2 1 1 ey ¼ 1 y þ y2 y3 þ R3 ; 2 6 where R3 ¼ y4 ec =24 where c is some number with 0 c 1. Since we have no better information as to the location of c, we must be content with the estimate 0 R3 y4 =24. Hence we have 1 1 2 ex ¼ 1 x2 þ x4 x6 þ R3 ; 2 6 where 0 R3 x8 =24, for x 2 ½0; 1. Therefore, we obtain Z 1 Z 1 Z 1 1 1 2 ex dx ¼ 1 x2 þ x4 x6 dx þ R3 dx 2 6 0 0 0 Z 1 1 1 1 R3 dx: ¼ 1 þ þ 3 10 42 0 R1 1 1 ¼ < 0:005, it follows that Since we have 0 0 R3 dx 9 24 216 Z 1 26 2 ð 0:7429Þ; ex dx 35 0 with an error less than 0.005. Equal Partitions If f : ½a; b ! R is continuous, we know that its Riemann integral exists. To find an approximate value for this integral with the minimum amount of calculation, it is convenient to consider partitions P n of ½a; b into n equal subintervals having length hn :¼ ðb aÞ=n. Hence P n is the partition: a < a þ hn < a þ 2hn < < a þ nhn ¼ b: If we pick our tag points to be the left endpoints and the right endpoints of the subintervals, we obtain the nth left approximation given by Ln ð f Þ :¼ hn n1 X k¼0 f ða þ khn Þ; C07 12/13/2010 10:5:43 Page 235 7.5 APPROXIMATE INTEGRATION 235 and the nth right approximation given by Rn ð f Þ :¼ hn n X f ða þ khn Þ: k¼1 It should be noted that it is almost as easy to evaluate both of these approximations as only one of them, since they differ only by the terms f(a) and f(b). Unless we have reason to believe that one of Ln ð f Þ or Rn ð f Þ is closer to the actual value of the integral than the other one, we generally take their mean: 1 ðLn ð f Þ þ Rn ð f ÞÞ; 2 which is readily seen to equal ð1Þ T n ð f Þ :¼ hn n1 X 1 1 f ð aÞ þ f ða þ khn Þ þ f ðbÞ ; 2 2 k¼1 Rb as a reasonable approximation to a f . However, we note that if f is increasing on [a, b], then it is clear from a sketch of the graph of f that Z b ð2Þ Ln ð f Þ f Rn ð f Þ: a In this case, we readily see that Z b 1 f T n ð f Þ ðRn ð f Þ Ln ð f ÞÞ 2 a 1 ð b aÞ : ¼ hn ð f ðbÞ f ðaÞÞ ¼ ð f ðbÞ f ðaÞÞ: 2 2n An error estimate such as this is useful, since it gives an upper bound for the error of the approximation in terms of quantities that are known at the outset. In particular, it can be used to determine how large we should choose n in order to have an approximation that will be correct to within a specified error e > 0. The above discussion was valid for the case that f is increasing on [a, b]. If f is decreasing, then the inequalities in (2) should be reversed. We can summarize both cases in the following statement. 7.5.1 Theorem If f : ½a; b ! R is monotone and if T n ð f Þ is given by (1), then Z b ð b aÞ : ð3Þ f T n ð f Þ j f ðbÞ f ðaÞj 2n a 7.5.2 Example IfRf ðxÞ :¼ ex on [0, 1], then f is decreasing. It follows from (3) that 2 1 ifR n ¼ 8, then j 0 ex dx T 8 ð f Þj ð1 e1 Þ=16 < 0:04, and if n ¼ 16, then 1 x2 j 0 e dx T 16 ð f Þj ð1 e1 Þ=32 < 0:02. Actually, the approximation is consider& ably better, as we will see in Example 7.5.5. 2 The Trapezoidal Rule The method of numerical integration called the ‘‘Trapezoidal Rule’’ is based on approximating the continuous function f : ½a; b ! R by a piecewise linear continuous function. C07 12/13/2010 10:5:44 236 Page 236 CHAPTER 7 THE RIEMANN INTEGRAL Let n 2 N and, as before, let hn :¼ ðb aÞ=n and consider the partition P n . We approximate f by the piecewise linear function gn that passes through the points Rb ða þ khn ; f ða þ khn ÞÞ, where k ¼ 0; 1; . . . ; n.RIt seems reasonable that the integral a f b will be ‘‘approximately equal to’’ the integral a gn when n is sufficiently large (provided that f is reasonably smooth). Since the area of a trapezoid with horizontal base h and vertical sides l 1 and l 2 is known to be 12 hðl 1 þ l 2 Þ, we have Z aþðkþ1Þhn 1 gn ¼ hn ½ f ða þ khn Þ þ f ða þ ðk þ 1Þhn Þ; 2 aþkhn for k ¼ 0; 1; . . . ; n 1. Summing these terms and noting that each partition point in P n except a and b belongs to two adjacent subintervals, we obtain Z b 1 1 gn ¼ hn f ðaÞ þ f ða þ hn Þ þ þ f ða þ ðk 1Þhn Þ þ f ðbÞ : 2 2 a But the term on the right is precisely T n ð f Þ, found in (1) as the mean of Ln ð f Þ and Rn ð f Þ. We call T n ð f Þ the nth Trapezoidal Approximation of f. In Theorem 7.5.1 we obtained an error estimate in the case where f is monotone; we now state one without this restriction on f, but in terms of the second derivative f 00 of f . 7.5.3 Theorem Let f, f 0 and f 00 be continuous on [a, b] and let T n ð f Þ be the nth Trapezoidal Approximation (1). Then there exists c 2 ½a; b such that Z b ðb aÞh2n 00 ð4Þ T nð f Þ f ðcÞ: f ¼ 12 a A proof of this result will be given in Appendix D; it depends on a number of results we have obtained in Chapters 5 and 6. The equality (4) is interesting in that it can give both an upper bound and a lower bound Rb for the difference T n ð f Þ a f . For example, if f 00 ðxÞ A > 0 for all x 2 ½a; b, then (4) 1 implies that this difference always exceeds 12 Aðb aÞh2n . If we only have f 00 ðxÞ 0 for x 2 ½a; b, which is the case when f is convex (¼ concave upward), then the Trapezoidal Approximation is always too large. The reader should draw a figure to visualize this. However, it is usually the upper bound that is of greater interest. 7.5.4 Corollary Let f, f 0 , and f 00 be continuous, and let j f 00 ðxÞj B2 for all x 2 ½a; b. Then Z b ðb aÞh2n ð b aÞ 3 B ð f Þ f ¼ B2 : ð5Þ T n 2 12 12n2 a When an upper bound B2 can be found, (5) can be used to determine how large n must be chosen in order to be certain of a desired accuracy. 7.5.5 Example If f ðxÞ :¼ ex on [0, 1], then a calculation shows that f 00 ðxÞ ¼ 2 2ex ð2x2 1Þ, so that we can take B2 ¼ 2. Thus, if n ¼ 8, then Z 1 2 1 T 8 ð f Þ ¼ < 0:003: f 12 64 384 0 2 C07 12/13/2010 10:5:44 Page 237 7.5 APPROXIMATE INTEGRATION 237 On the other hand, if n ¼ 16, then we have Z 1 2 1 T 16 ð f Þ ¼ < 0:000 66: f 12 256 1 536 0 Thus, the accuracy in this case is considerably better than predicted in Example & 7.5.2. The Midpoint Rule One obvious method of approximating the integral of f is to take the Riemann sums evaluated at the midpoints of the subintervals. Thus, if P n is the equally spaced partition given before, the Midpoint Approximation of f is given by 1 3 1 M h h ð f Þ :¼ h f a þ hn þ f a þ þ þ f a n ð6Þ n n n n 2 2 2 n X 1 ¼ hn f a þ k hn : 2 k¼1 Another method might be to use piecewise linear functions that are tangent to the graph of f at the midpoints of these subintervals. At first glance, it seems as if we would need to know the slope of the tangent line to the graph of f at each of the midpoints a þ k 12 hn ðk ¼ 1; 2; . . . ; nÞ. However, it is an exercise in geometry to1show that the area of the trapezoid whose top is this tangent line at the midpoint a þ k 2 hn is equal to the area of the rectangle whose height is f a þ k 12 hn . (See Figure 7.5.1.) Thus, this area is given by (6), and the ‘‘Tangent Trapezoid Rule’’ turns out to be the same as the ‘‘Midpoint Rule.’’ We now state a theorem showing that the Midpoint Rule gives better accuracy than the Trapezoidal Rule by a factor of 2. 7.5.6 Theorem Let f, f 0 , and f 00 be continuous on [a, b] and let M n ð f Þ be the nth Midpoint Approximation (6). Then there exists g 2 ½a; b such that Z b ðb aÞh2n 00 ð7Þ f ðg Þ: f Mnð f Þ ¼ 24 a The proof of this result is in Appendix D. Figure 7.5.1 The tangent trapezoid C07 12/13/2010 10:5:45 238 Page 238 CHAPTER 7 THE RIEMANN INTEGRAL As in the case with Theorem 7.5.3,Rformula (7) can be used to give both an upper bound b and a lower bound for the difference a f M n ð f Þ, although it is an upper bound that is usually of greater interest. In contrast with the Trapezoidal Rule, if the function is convex, then the Midpoint Approximation is always too small. The next result is parallel to Corollary 7.5.4. 7.5.7 Corollary Let f, f 0 , and f 00 be continuous, and let j f 00 ðxÞj B2 for all x 2 ½a; b. Then Z b ðb aÞh2n ð b aÞ 3 ð8Þ M B ð f Þ f ¼ B2 : n 2 24 24n2 a Simpson’s Rule The final approximation procedure that we will consider usually gives a better approximation than either the Trapezoidal or the Midpoint Rule and requires essentially no extra calculation. However, the convexity (or the concavity) of f does not give any information about the error for this method. Whereas the Trapezoidal and Midpoint Rules were based on the approximation of f by piecewise linear functions, Simpson’s Rule approximates the graph of f by parabolic arcs. To help motivate the formula, the reader may show that if three points ðh; y0 Þ; ð0; y1 Þ; and ðh; y2 Þ are given, then the quadratic function qðxÞ :¼ Ax2 þ Bx þ C that passes through these points has the property that Z h 1 q ¼ hðy0 þ 4y1 þ y2 Þ: 3 h Now let f be a continuous function on [a, b] and let n 2 N be even, and let hn :¼ ðb aÞ=n. On each ‘‘double subinterval’’ ½a; a þ 2hn ; ½a þ 2hn ; a þ 4hn ; . . . ; ½b 2hn ; b; we approximate f by n=2 quadratic functions that agree with f at the points y0 :¼ f ðaÞ; y1 :¼ f ða þ hn Þ; y2 :¼ f ða þ 2hn Þ; . . . ; yn :¼ f ðbÞ: These considerations lead to the nth Simpson Approximation, defined by ð9Þ Sn ð f Þ :¼ 1 hn f ðaÞ þ 4f ða þ hn Þ þ 2 f ða þ 2hn Þ þ 4 f ða þ 3hn Þ 3 þ 2 f ða þ 4hn Þ þ þ 2 f ðb 2hn Þ þ 4 f ðb hn Þ þ f ðbÞ : Note that the coefficients of the values of f at the n þ 1 partition points follow the pattern 1; 4; 2; 4; 2; . . . ; 4; 2; 4; 1. We now state a theorem that gives an estimate about the accuracy of the Simpson Approximation; it involves the fourth derivative of f. 7.5.8 Theorem Let f, f 0 , f 00 , f ð3Þ , and f ð4Þ be continuous on [a, b] and let n 2 N be even. If Sn ð f Þ is the nth Simpson Approximation (9), then there exists c 2 ½a; b such that Z b ðb aÞh4n ð4Þ f ðcÞ: ð10Þ Sn ð f Þ f ¼ 180 a A proof of this result is given in Appendix D. C07 12/13/2010 10:5:46 Page 239 7.5 APPROXIMATE INTEGRATION 239 The next result is parallel to Corollaries 7.5.4 and 7.5.7. 7.5.9 Corollary Let f, f 0 , f 00 , f ð3Þ , and f ð4Þ be continuous on [a, b] and let f ð4Þ ðxÞ B4 for all x 2 ½a; b. Then Z b ðb aÞh4n ð b aÞ 5 Sn ð f Þ ð11Þ B f ¼ B4 : 4 180 180n4 a Successful use of the estimate (11) depends on being able to find an upper bound for the fourth derivative. 7.5.10 Example If f ðxÞ :¼ 4ex on [0, 1] then a calculation shows that 2 f ð4Þ ðxÞ ¼ 4ex 4x4 12x2 þ 3 ; whence it follows that f ð4Þ ðxÞ 20 for x 2 ½0; 1, so we can take B4 ¼ 20. It follows from (11) that if n ¼ 8 then Z 1 1 1 S8 ð f Þ < 0:000 03 f 20 ¼ 36; 864 180 84 0 2 and that if n ¼ 16 then Z S16 ð f Þ 1 0 1 < 0:000 001 7: f 589;824 & Remark The nth Midpoint Approximation M n ð f Þ can be used to ‘‘step up’’ to the (2n)th Trapezoidal and Simpson Approximations by using the formulas 1 1 T 2n ð f Þ ¼ M n ð f Þ þ T n ð f Þ 2 2 and 2 1 S2n ð f Þ ¼ M n ð f Þ þ T n ð f Þ; 3 3 that are given in the exercises. Thus once the initial Trapezoidal Approximation T 1 ¼ T 1 ð f Þ has been calculated, only the Midpoint Approximations M n ¼ M n ð f Þ need be found. That is, we employ the following sequence of calculations: 1 T 1 ¼ ðb aÞð f ðaÞ þ f ðbÞÞ; 2 M 1 ¼ ðb aÞf 1 ða þ bÞ ; 2 M2 ; M4 ; 1 1 M1 þ T 1 ; 2 2 1 1 T 4 ¼ M2 þ T 2 ; 2 2 1 T 8 ¼ M4 þ 1 T 4 ; 2 2 T2 ¼ 2 1 M1 þ T 1; 3 3 2 1 S4 ¼ M 2 þ T 2 ; 3 3 2 S8 ¼ M 4 þ 1 T 4 ; 3 3 S2 ¼ Exercises for Section 7.5 1. Use the Trapezoidal Approximation with n ¼ 4 to evaluate ln 2 ¼ 0:6866 ln 2 0:6958 and that 0:0013 < 1 1 T 4 ln 2 < 0:0105: 768 96 R2 1 ð1=xÞdx. Show that C07 12/13/2010 10:5:46 Page 240 240 CHAPTER 7 THE RIEMANN INTEGRAL 2. Use the Simpson Approximation with n ¼ 4 to evaluate ln 2 ¼ ln 2 0:6933 and that 0:000 016 < R2 1 ð1=xÞdx. Show that 0:6927 1 1 1 S4 ln 2 < 0:000 521: 5 1920 1920 2 1 3 for x 2 ½0; 1. Show that f 00 ðxÞ ¼ 2ð3x2 1Þð1 þ x2 Þ and that 3. Let f ðxÞ :¼ ð1 þ x2 Þ 00 j f ðxÞj 2 for x 2 [0, 1]. Use the Trapezoidal Approximation with n ¼ 4 to evaluate p=4 ¼ R1 0 f ðxÞdx. Show that jT 4 ð f Þ ðp=4Þj 1=96 < 0:0105: 4. If the Trapezoidal Approximation T n ð f Þ is used to approximate p=4 as in Exercise 3, show that we must take n 409 in order to be sure that the error is less than 106 . 5 5. Let f be as in Exercise 3. Show that f ð4Þ ðxÞ ¼ 24ð5x4 10x2 þ 1Þð1 þ x2 Þ and that ð4Þ f ðxÞ 96 for x 2 ½0; 1. Use Simpson’s Approximation with n ¼ 4 to evaluate p=4. Show that jS4 ð f Þ ðp=4Þj 1=480 < 0:0021: 6. If the Simpson Approximation Sn ð f Þ is used to approximate p=4 as in Exercise 5, show that we must take n 28 in order to be sure that the error is less than 106 . 7. If p is a polynomial of degree at most 3, show that the Simpson Approximations are exact. 8. Show that if f 00 ðxÞ 0 on ½a; b (that is, if f is convex on [a, b]), then for any natural numbers Rb m, n we have M n ð f Þ a f ðxÞdx T m ð f Þ: If f 00 ðxÞ 0 on [a, b], this inequality is reversed. 9. Show that T 2n ð f Þ ¼ 12 ½M n ð f Þ þ T n ð f Þ: 10. Show that S2n ð f Þ ¼ 23 M n ð f Þ þ 13 T n ð f Þ: h i Rb 11. Show that one has the estimate Sn ð f Þ a f ðxÞdx ðb aÞ2 =18n2 B2 , where B2 j f 00 ðxÞj for all x 2 ½a; b. R1 1=2 12. Note that 0 ð1 x2 Þ dx ¼ p=4. Explain why the error estimates given by formulas (4), (7), 1=2 for x in [0, 1], then and (10) cannot be used. Show that if hðxÞ ¼ ð1 x2 Þ T n ðhÞ p=4 M n ðhÞ. Calculate M 8 ðhÞ and T 8 ðhÞ. R 1=pffiffi2 13. If h is as in Exercise 12, explain why K :¼ 0 hðxÞdx ¼ p=8 þ 1=4. Show that jh00 ðxÞj p ffiffi ffi 23=2 and that hð4Þ ðxÞ 9 27=2 for x 2 0; 1= 2 . Show that jK T n ðhÞj 1=12n2 and that jK Sn ðhÞj 1=10n4 . Use these results to calculate p. In Exercises 14–20, approximate the indicated integrals, giving estimates for the error. Use a calculator to obtain a high degree of precision. Z 2 14. 1 þ x4 1=2 Z dx: 2 15. 0 4 þ x3 1=2 Z dx: 16. 0 Z p 17. 0 Z 1 20. 0 sin x dx: x cos x2 dx: Z 18. 0 p=2 dx : 1 þ sin x Z 0 19. 0 1 dx : 1 þ x3 p=2 pffiffiffiffiffiffiffiffiffiffi sin x dx: C08 12/08/2010 14:48:14 Page 241 CHAPTER 8 SEQUENCES OF FUNCTIONS In previous chapters we have often made use of sequences of real numbers. In this chapter we shall consider sequences whose terms are functions rather than real numbers. Sequences of functions arise naturally in real analysis and are especially useful in obtaining approximations to a given function and defining new functions from known ones. In Section 8.1 we will introduce two different notions of convergence for a sequence of functions: pointwise convergence and uniform convergence. The latter type of convergence is very important, and will be the main focus of our attention. The reason for this focus is the fact that, as is shown in Section 8.2, uniform convergence ‘‘preserves’’ certain properties in the sense that if each term of a uniformly convergent sequence of functions possesses these properties, then the limit function also possesses the properties. In Section 8.3 we will apply the concept of uniform convergence to define and derive the basic properties of the exponential and logarithmic functions. Section 8.4 is devoted to a similar treatment of the trigonometric functions. Section 8.1 Pointwise and Uniform Convergence Let A R be given and suppose that for each n 2 N there is a function f n : A ! R; we shall say that ( fn) is a sequence of functions on A to R. Clearly, for each x 2 A, such a sequence gives rise to a sequence of real numbers, namely the sequence ð1Þ ð f n ðxÞÞ; obtained by evaluating each of the functions at the point x. For certain values of x 2 A the sequence (1) may converge, and for other values of x 2 A this sequence may diverge. For each x 2 A for which the sequence (1) converges, there is a uniquely determined real number limð f n ðxÞÞ. In general, the value of this limit, when it exists, will depend on the choice of the point x 2 A. Thus, there arises in this way a function whose domain consists of all numbers x 2 A for which the sequence (1) converges. 8.1.1 Definition Let ð f n Þ be a sequence of functions on A R to R, let A0 A, and let f : A0 ! R. We say that the sequence ð f n Þ converges on A0 to f if, for each x 2 A0 , the sequence ð f n ðxÞÞ converges to f (x) in R. In this case we call f the limit on A0 of the sequence ð f n Þ. When such a function f exists, we say that the sequence ð f n Þ is convergent on A0, or that ð f n Þ converges pointwise on A0. It follows from Theorem 3.1.4 that, except for a possible modification of the domain A0, the limit function is uniquely determined. Ordinarily we choose A0 to be the largest set possible; that is, we take A0 to be the set of all x 2 A for which the sequence (1) is convergent in R. 241 C08 12/08/2010 14:48:15 242 Page 242 CHAPTER 8 SEQUENCES OF FUNCTIONS In order to symbolize that the sequence ð f n Þ converges on A0 to f, we sometimes write f ¼ limð f n Þ on A0 ; or f n ! f on A0 : Sometimes, when fn and f are given by formulas, we write f ðxÞ ¼ lim f n ðxÞ for x 2 A0 ; or f n ðxÞ ! f ðxÞ for x 2 A0 : 8.1.2 Examples (a) lim(x=n) ¼ 0 for x 2 R. For n 2 N, let f n ðxÞ :¼ x=n and let f ðxÞ :¼ 0 for x 2 R. By Example 3.1.6(a), we have lim(1=n) ¼ 0. Hence it follows from Theorem 3.2.3 that limð f n ðxÞÞ ¼ limðx=nÞ ¼ x limð1=nÞ ¼ x 0 ¼ 0 for all x 2 R. (See Figure 8.1.1.) Figure 8.1.1 fn(x) ¼ x=n Figure 8.1.2 gn(x) ¼ xn (b) lim(xn). Let gn ðxÞ :¼ xn for x 2 R; n 2 N. (See Figure 8.1.2.) Clearly, if x ¼ 1, then the sequence (gn(1)) ¼ (1) converges to 1. It follows from Example 3.1.11(b) that lim(xn) ¼ 0 for 0 x < 1 and it is readily seen that this is also true for 1 < x < 0. If x ¼ 1, then gn ð1Þ ¼ ð1Þn , and it was seen in Example 3.2.8(b) that the sequence is divergent. Similarly, if jxj > 1, then the sequence (xn) is not bounded, and so it is not convergent in R. We conclude that if 0 for 1 < x < 1; gðxÞ :¼ 1 for x ¼ 1; then the sequence (gn) converges to g on the set ð1; 1. (c) limððx2 þ nxÞ=nÞ ¼ x for x 2 R. Let hn ðxÞ: ¼ ðx2 þ nxÞ=n for x 2 R, n 2 N, and let hðxÞ :¼ x for x 2 R. (See Figure 8.1.3.) Since we have hn ðxÞ ¼ ðx2 =nÞ þ x, it follows from Example 3.1.6(a) and Theorem 3.2.3 that hn ðxÞ ! x ¼ hðxÞ for all x 2 R. C08 12/08/2010 14:48:16 Page 243 8.1 POINTWISE AND UNIFORM CONVERGENCE Figure 8.1.3 hn (x) ¼ (x2 þ nx)=n Figure 8.1.4 243 Fn(x) ¼ sin(nx þ n)=n (d) lim((1=n) sin(nx þ n)) ¼ 0 for x 2 R. Let F n ðxÞ :¼ ð1=nÞ sinðnx þ nÞ for x 2 R; n 2 N; and let FðxÞ :¼ 0 for x 2 R. (See Figure 8.1.4.) Since j sin y j 1 for all y 2 R we have 1 1 ð2Þ j F n ðxÞ FðxÞj ¼ sinðnx þ nÞ n n for all x 2 R. Therefore it follows that lim(Fn(x)) ¼ 0 ¼ F(x) for all x 2 R. The reader should note that, given any e > 0, if n is sufficiently large, then jF n ðxÞ FðxÞj < e for all & values of x simultaneously! Partly to reinforce Definition 8.1.1 and partly to prepare the way for the important notion of uniform convergence, we reformulate Definition 8.1.1 as follows. 8.1.3 Lemma A sequence ( fn) of functions on A R to R converges to a function f : A0 ! R on A0 if and only if for each e > 0 and each x 2 A0 there is a natural number Kðe; xÞ such that if n Kðe; xÞ, then ð3Þ j f n ðxÞ f ðxÞj < e: We leave it to the reader to show that this is equivalent to Definition 8.1.1. We wish to emphasize that the value of Kðe; xÞ will depend, in general, on both e > 0 and x 2 A0 . The reader should confirm the fact that in Examples 8.1.2(a–c), the value of Kðe; xÞ required to obtain an inequality such as (3) does depend on both e > 0 and x 2 A0 . The intuitive reason for this is that the convergence of the sequence is ‘‘significantly faster’’ at some points than it is at others. However, in Example 8.1.2(d), as we have seen in inequality (2), if we choose n sufficiently large, we can make jF n ðxÞ FðxÞj < e for all values of x 2 R. It is precisely this rather subtle difference that distinguishes between the notion of the ‘‘pointwise convergence’’ of a sequence of functions (as defined in Definition 8.1.1) and the notion of ‘‘uniform convergence.’’ Uniform Convergence 8.1.4 Definition A sequence ( fn) of functions on A R to R converges uniformly on A0 A to a function f : A0 ! R if for each e > 0 there is a natural number KðeÞ (depending on e but not on x 2 A0 ) such that if n KðeÞ, then ð4Þ j f n ðxÞ f ðxÞj < e for all x 2 A0 : C08 12/08/2010 14:48:16 244 Page 244 CHAPTER 8 SEQUENCES OF FUNCTIONS In this case we say that the sequence ( fn) is uniformly convergent on A0. Sometimes we write fn F f on A0 ; f n ðxÞ F or f ðxÞ for x 2 A0 : It is an immediate consequence of the definitions that if the sequence ( fn) is uniformly convergent on A0 to f, then this sequence also converges pointwise on A0 to f in the sense of Definition 8.1.1. That the converse is not always true is seen by a careful examination of Examples 8.1.2(a–c); other examples will be given below. It is sometimes useful to have the following necessary and sufficient condition for a sequence ( fn) to fail to converge uniformly on A0 to f. 8.1.5 Lemma A sequence ( fn) of functions on A R to R does not converge uniformly on A0 A to a function f : A0 ! R if and only if for some e0 > 0 there is a subsequence f nk of ð f n Þ and a sequence ðxk Þ in A0 such that f n ðxk Þ f ðxk Þ e0 k ð5Þ for all k 2 N: The proof of this result requires only that the reader negate Definition 8.1.4; we leave this to the reader as an important exercise. We now show how this result can be used. 8.1.6 Examples (a) 8.1.2(a). If we let nk :¼ k and xk :¼ k, then Consider Example f nk ðxk Þ ¼ 1 so that f nk ðxk Þ f ðxk Þ ¼ j1 0j ¼ 1. Therefore the sequence ( fn) does not converge uniformly on R to f. 1=k , then (b) Consider Example 8.1.2(b). If nk :¼ k and xk :¼ 12 1 1 gn ðxk Þ gðxk Þ ¼ 0 ¼ : 2 2 k Therefore the sequence (gn) does not converge uniformly on ð1; 1 to g. (c) Consider Example 8.1.2(c). If nk :¼ k and xk :¼ k, then hnk ðxk Þ ¼ 0 and hðxk Þ ¼ k so that jhnk ðxk Þ hðxk Þj ¼ k. Therefore the sequence (hn) does not converge uni& formly on R to h. The Uniform Norm In discussing uniform convergence, it is often convenient to use the notion of the uniform norm on a set of bounded functions. 8.1.7 Definition If A R and w : A ! R is a function, we say that w is bounded on A if the set w(A) is a bounded subset of R. If w is bounded we define the uniform norm of w on A by ð6Þ jj wjjA :¼ supfjwðxÞj : x 2 Ag: Note that it follows that if e > 0, then ð7Þ jj wjjA e () jwðxÞj e for all x 2 A: 8.1.8 Lemma A sequence ( fn) of bounded functions on A R converges uniformly on A to f if and only if jj f n f jjA ! 0. C08 12/08/2010 14:48:17 Page 245 8.1 POINTWISE AND UNIFORM CONVERGENCE 245 Proof. ()) If ( fn) converges uniformly on A to f, then by Definition 8.1.4, given any e > 0 there exists K(e) such that if n KðeÞ and x 2 A then j f n ðxÞ f ðxÞj e: From the definition of supremum, it follows that jj f n f jjA e whenever n KðeÞ. Since e > 0 is arbitrary this implies that jj f n f jjA ! 0. (() If jj f n f jjA ! 0, then given e > 0 there is a natural number H(e) such that if n HðeÞ then jj f n f jjA e. It follows from (7) that j f n ðxÞ f ðxÞj e for all n HðeÞ and x 2 A. Therefore ( fn) converges uniformly on A to f. Q.E.D. We now illustrate the use of Lemma 8.1.8 as a tool in examining a sequence of bounded functions for uniform convergence. 8.1.9 Examples (a) We cannot apply Lemma 8.1.8 to the sequence in Example 8.1.2(a) since the function f n ðxÞ f ðxÞ ¼ x=n is not bounded on R. For the sake of illustration, let A :¼ ½0; 1. Although the sequence (x=n) did not converge uniformly on R to the zero function, we shall show that the convergence is uniform on A. To see this, we observe that 1 jj f n f jjA ¼ supfjx=n 0j : 0 x 1g ¼ n so that jj f n f jjA ! 0. Therefore ( fn) is uniformly convergent on A to f. (b) Let gn ðxÞ :¼ xn for x 2 A :¼ ½0; 1 and n 2 N, and let gðxÞ :¼ 0 for 0 x < 1 and gð1Þ :¼ 1. The functions gn ðxÞ gðxÞ are bounded on A and n for 0 x < 1 x ¼1 jj gn gjjA ¼ sup 0 for x ¼ 1 for any n 2 N. Since jj gn gjjA does not converge to 0, we infer that the sequence (gn) does not converge uniformly on A to g. (c) We cannot apply Lemma 8.1.8 to the sequence in Example 8.1.2(c) since the function hn ðxÞ hðxÞ ¼ x2 =n is not bounded on R. Instead, let A :¼ ½0; 8 and consider jj hn hjjA ¼ sup x2 =n : 0 x 8 ¼ 64=n: Therefore, the sequence (hn) converges uniformly on A to h. (d) If we refer to Example 8.1.2(d), we see from (2) that jj F n F jjR 1=n. Hence (Fn) converges uniformly on R to F. (e) Let GðxÞ :¼ xn ð1 xÞ for x 2 A :¼ ½0; 1. Then the sequence (Gn(x)) converges to GðxÞ :¼ 0 for each x 2 A. To calculate the uniform norm of Gn G ¼ Gn on A, we find the derivative and solve G0n ðxÞ ¼ xn1 ðn ðn þ 1ÞxÞ ¼ 0 to obtain the point xn :¼ n=ðn þ 1Þ. This is an interior point of [0, 1], and it is easily verified by using the First Derivative Test 6.2.8 that Gn attains a maximum on [0, 1] at xn. Therefore, we obtain 1 ; jj Gn jjA ¼ Gn ðxn Þ ¼ ð1 þ 1=nÞn nþ1 which converges to ð1=eÞ 0 ¼ 0. Thus we see that convergence is uniform on A. & C08 12/08/2010 14:48:17 246 Page 246 CHAPTER 8 SEQUENCES OF FUNCTIONS By making use of the uniform norm, we can obtain a necessary and sufficient condition for uniform convergence that is often useful. 8.1.10 Cauchy Criterion for Uniform Convergence Let ( fn) be a sequence of bounded functions on A R. Then this sequence converges uniformly on A to a bounded function f if and only if for each e > 0 there is a number H(e) in N such that for all m; n HðeÞ, then & jj f m f n jjA e. Proof. ()) If fn F f on A, then given e > 0 there exists a natural number K 12 e such that if n K 12 e then jj f n f jjA 12 e. Hence, if both m; n K 12 e , then we conclude that j f m ðxÞ f n ðxÞj j f m ðxÞ f ðxÞj þ j f n ðxÞ f ðxÞj 12 e þ 12 e ¼ e for all x 2 A. Therefore jj f m f n jjA e for m; n K 12 e ¼: HðeÞ. (() Conversely, suppose that for e > 0 there is H(e) such that if m; n HðeÞ, then jj f m f n jjA e. Therefore, for each x 2 A we have ð8Þ j f m ðxÞ f n ðxÞj jj f m f n jjA e for m; n HðeÞ: It follows that (fn(x)) is a Cauchy sequence in R; therefore, by Theorem 3.5.5, it is a convergent sequence. We define f : A ! R by f ðxÞ :¼ limð f n ðxÞÞ for x 2 A: If we let n ! 1 in (8), it follows from Theorem 3.2.6 that for each x 2 A we have j f m ðxÞ f ðxÞj e for m HðeÞ: Therefore the sequence ( fn) converges uniformly on A to f. Q.E.D. Exercises for Section 8.1 1. Show that lim(x=(x þ n)) ¼ 0 for all x 2 R; x 0. 2. Show that lim(nx=(1 þ n2x2)) ¼ 0 for all x 2 R. 3. Evaluate lim(nx=(1 þ nx)) for x 2 R; x 0. 4. Evaluate lim(xn=(1 þ xn)) for x 2 R; x 0. 5. Evaluate lim((sin nx)=(1 þ nx)) for x 2 R; x 0. 6. Show that lim(Arctan nx) ¼ (p=2)sgn x for x 2 R. 7. Evaluate lim(enx ) for x 2 R; x 0. 8. Show that limðxenx Þ ¼ 0 for x 2 R; x 0. 9. Show that limðx2 enx Þ ¼ 0 and that limðn2 x2 enx Þ ¼ 0 for x 2 R; x 0. 10. Show that limððcos pxÞ2n Þ exists for all x 2 R. What is its limit? 11. Show that if a > 0, then the convergence of the sequence in Exercise 1 is uniform on the interval [0, a], but is not uniform on the interval [0, 1). 12. Show that if a > 0, then the convergence of the sequence in Exercise 2 is uniform on the interval [a, 1), but is not uniform on the interval [0, 1). 13. Show that if a > 0, then the convergence of the sequence in Exercise 3 is uniform on the interval [a, 1), but is not uniform on the interval [0, 1). C08 12/08/2010 14:48:17 Page 247 8.2 INTERCHANGE OF LIMITS 247 14. Show that if 0 < b < 1, then the convergence of the sequence in Exercise 4 is uniform on the interval [0, b], but is not uniform on the interval [0, 1]. 15. Show that if a > 0, then the convergence of the sequence in Exercise 5 is uniform on the interval [a, 1), but is not uniform on the interval [0, 1). 16. Show that if a > 0, then the convergence of the sequence in Exercise 6 is uniform on the interval [a, 1), but is not uniform on the interval (0, 1). 17. Show that if a > 0, then the convergence of the sequence in Exercise 7 is uniform on the interval [a, 1), but is not uniform on the interval [0, 1). 18. Show that the convergence of the sequence in Exercise 8 is uniform on [0, 1). 19. Show that the sequence ðx2 enx Þ converges uniformly on [0, 1). 20. Show that if a > 0, then the sequence ðn2 x2 enx Þ converges uniformly on the interval [a, 1), but that it does not converge uniformly on the interval [0, 1). 21. Show that if ( fn), (gn) converge uniformly on the set A to f, g, respectively, then ( fn þ gn) converges uniformly on A to f þ g. 22. Show that if f n ðxÞ :¼ x þ 1=n and f ðxÞ :¼ x for x 2 R, then ( fn) converges uniformly on R to f, but the sequence f 2n does not converge uniformly on R. (Thus the product of uniformly convergent sequences of functions may not converge uniformly.) 23. Let ( fn), (gn) be sequences of bounded functions on A that converge uniformly on A to f, g, respectively. Show that ð f n gn Þ converges uniformly on A to fg. 24. Let ( fn) be a sequence of functions that converges uniformly to f on A and that satisfies j f n ðxÞj M for all n 2 N and all x 2 A. If g is continuous on the interval ½M; M , show that the sequence (g fn) converges uniformly to g f on A. Section 8.2 Interchange of Limits It is often useful to know whether the limit of a sequence of functions is a continuous function, a differentiable function, or a Riemann integrable function. Unfortunately, it is not always the case that the limit of a sequence of functions possesses these useful properties. 8.2.1 Examples (a) Let gn ðxÞ :¼ xn for x 2 ½0; 1 and n 2 N. Then, as we have noted in Example 8.1.2(b), the sequence (gn) converges pointwise to the function 0 for 0 x < 1; gðxÞ :¼ 1 for x ¼ 1: Although all of the functions gn are continuous at x ¼ 1, the limit function g is not continuous at x ¼ 1. Recall that it was shown in Example 8.1.6(b) that this sequence does not converge uniformly to g on [0, 1]. (b) Each of the functions gn ðxÞ ¼ xn in part (a) has a continuous derivative on [0,1]. However, the limit function g does not have a derivative at x ¼ 1, since it is not continuous at that point. (c) Let f n : ½0; 1 ! R be defined for n 2 by 8 <n2 x f n ðxÞ :¼ n2 ðx 2=nÞ : 0 for for for 0 x 1=n; 1=n x 2=n; 2=n x 1: C08 12/08/2010 14:48:17 248 Page 248 CHAPTER 8 SEQUENCES OF FUNCTIONS (See Figure 8.2.1.) It is clear that each of the functions fn is continuous on [0, 1]; hence it is Riemann integrable. Either by means of a direct calculation, or by referring to the significance of the integral as an area, we obtain Z 1 f n ðxÞdx ¼ 1 for n 2: 0 The reader may show that f n ðxÞ ! 0 for all x 2 ½0; 1; hence R 1the limit function f vanishes identically and is continuous (and hence integrable), and 0 f ðxÞdx ¼ 0. Therefore we have the uncomfortable situation that: Z 1 Z 1 f ðxÞdx ¼ 0 6¼ 1 ¼ lim f n ðxÞdx: 0 0 Figure 8.2.1 Example 8.2.1(c) (d) Those who consider the functions fn in part (c) to be ‘‘artificial’’ may prefer to consider 2 the sequence (hn) defined by hn ðxÞ :¼ 2nxenx for x 2 ½0; 1; n 2 N. Since hn ¼ H 0 n , 2 where H n ðxÞ :¼ enx , the Fundamental Theorem 7.3.1 gives Z 1 hn ðxÞdx ¼ H n ð1Þ H n ð0Þ ¼ 1 en : 0 It is an exercise to show that hðxÞ :¼ limðhn ðxÞÞ ¼ 0 for all x 2 ½0; 1; hence Z 1 Z 1 hðxÞdx 6¼ lim 0 hn ðxÞdx: & 0 Although the extent of the discontinuity of the limit function in Example 8.2.1 (a) is not very great, it is evident that more complicated examples can be constructed that will produce more extensive discontinuity. In any case, we must abandon the hope that the limit of a convergent sequence of continuous [respectively, differentiable, integrable] functions will be continuous [respectively, differentiable, integrable]. It will now be seen that the additional hypothesis of uniform convergence is sufficient to guarantee that the limit of a sequence of continuous functions is continuous. Similar results will also be established for sequences of differentiable and integrable functions. Interchange of Limit and Continuity 8.2.2 Theorem Let ( fn) be a sequence of continuous functions on a set A R and suppose that ( fn) converges uniformly on A to a function f : A ! R. Then f is continuous on A. C08 12/08/2010 14:48:18 Page 249 8.2 INTERCHANGE OF LIMITS 249 Proof. By hypothesis, given e > 0 there exists a natural number H :¼ H 13 e such that if n H then j f n ðxÞ f ðxÞj < 13 e for all x 2 A. Let c 2 A be arbitrary; we will show that f is continuous at c. By the Triangle Inequality we have j f ðxÞ f ðcÞj j f ðxÞ f H ðxÞj þ j f H ðxÞ f H ðcÞj þ j f H ðcÞ f ðcÞj 13 e þ j f H ðxÞ f H ðcÞj þ 13 e: Since fH is continuous at c, there exists a number d :¼ d 13 e; c; f H > 0 such that if jx cj < d and x 2 A, then j f H ðxÞ f H ðcÞj < 13 e. Therefore, if jx cj < d and x 2 A, then we have j f ðxÞ f ðcÞj < e. Since e > 0 is arbitrary, this establishes the continuity of f at the arbitrary point c 2 A. (See Figure 8.2.2.) Q.E.D. Figure 8.2.2 j f ðxÞ f ðcÞj < e Remark Although the uniform convergence of the sequence of continuous functions is sufficient to guarantee the continuity of the limit function, it is not necessary. (See Exercise 2.) Interchange of Limit and Derivative We mentioned in Section 6.1 that Weierstrass showed that the function defined by the series 1 X f ðxÞ :¼ 2k cos 3k x k¼0 is continuous at every point but does not have a derivative at any point in R. By considering the partial sums of this series, we obtain a sequence of functions ( fn) that possess a derivative at every point and are uniformly convergent to f. Thus, even though the sequence of differentiable functions ( fn) is uniformly convergent, it does not follow that the limit function is differentiable. (See Exercises 9 and 10.) We now show that if the sequence of derivatives f 0n is uniformly convergent, then all is well. If one adds the hypothesis that the derivatives are continuous, then it is possible to give a short proof, based on the integral. (See Exercise 11.) However, if the derivatives are not assumed to be continuous, a somewhat more delicate argument is required. 8.2.3 Theorem Let J R be a bounded interval and let ( fn) be a sequence of functions on J to R. Suppose that there exists x0 2 J such that ð f n ðx0 ÞÞ converges, and that the sequence f 0n of derivatives exists on J and converges uniformly on J to a function g. Then the sequence ( fn) converges uniformly on J to a function f that has a derivative at every point of J and f 0 ¼ g. C08 12/08/2010 14:48:18 250 Page 250 CHAPTER 8 SEQUENCES OF FUNCTIONS Proof. Let a < b be the endpoints of J and let x 2 J be arbitrary. If m; n 2 N, we apply the Mean Value Theorem 6.2.4 to the difference fm fn on the interval with endpoints x0, x. We conclude that there exists a point y (depending on m, n) such that f m ðxÞ f n ðxÞ ¼ f m ðx0 Þ f n ðx0 Þ þ ðx x0 Þf f m0 ðyÞ f n0 ðyÞg: Hence we have ð1Þ jj f m f n jjJ j f m ðx0 Þ f n ðx0 Þj þ ðb aÞjj f m0 f n0 jjJ : From Theorem 8.1.10, it follows from (1) and the hypotheses that ð f n ðx0 ÞÞ is convergent and that ð f n0 Þ is uniformly convergent on J, that ( fn) is uniformly convergent on J. We denote the limit of the sequence ( fn) by f. Since the fn are all continuous and the convergence is uniform, it follows from Theorem 8.2.2 that f is continuous on J. To establish the existence of the derivative of f at a point c 2 J, we apply the Mean Value Theorem 6.2.4 to f m f n on an interval with end points c, x. We conclude that there exists a point z (depending on m, n) such that f f m ðxÞ f n ðxÞg f f m ðcÞ f n ðcÞg ¼ ðx cÞf f m0 ðzÞ f n0 ðzÞg: Hence, if x 6¼ c, we have f m ðxÞ f m ðcÞ f n ðxÞ f n ðcÞ jj f m0 f n0 jj : J xc xc Since ð f n0 Þ converges uniformly on J, if e > 0 is given there exists H(e) such that if m; n HðeÞ and x 6¼ c, then f m ðxÞ f m ðcÞ f n ðxÞ f n ðcÞ e: ð2Þ xc xc If we take the limit in (2) with respect to m and use Theorem 3.2.6, we have f ðxÞ f ðcÞ f n ðxÞ f n ðcÞ e: xc xc 0 provided that x 6¼ c, n H(e). Since gðcÞ ¼ lim f n ðcÞ , there exists N(e) such that if n N(e), then j f n0 ðcÞ gðcÞj < e. Now let K :¼ supfHðeÞ; NðeÞg. Since f K0 ðcÞ exists, there exists dK(e) > 0 such that if 0 < j x cj < dK ðeÞ, then f K ðxÞ f K ðcÞ 0 < e: f ðcÞ K xc Combining these inequalities, we conclude that if 0 < j x cj < dK ðeÞ, then f ðxÞ f ðcÞ x c gðcÞ < 3e: Since e > 0 is arbitrary, this shows that f 0(c) exists and equals g(c). Since c 2 J is arbitrary, we conclude that f 0¼ g on J. Q.E.D. Interchange of Limit and Integral We have seen in Example 8.2.1(c) that if ( fn) is a sequence R½a; b that converges on [a, b] to a function f in R½a; b, then it need not happen that Z b Z b ð3Þ f ¼ lim f n: a n!1 a C08 12/08/2010 14:48:19 Page 251 8.2 INTERCHANGE OF LIMITS 251 We will now show that uniform convergence of the sequence is sufficient to guarantee that this equality holds. 8.2.4 Theorem Let ( fn) be a sequence of functions in R½a; b and suppose that ( fn) converges uniformly on [a, b] to f. Then f 2 R½a; b and (3) holds. Proof. It follows from the Cauchy Criterion 8.1.10 that given e > 0 there exists H(e) such that if m > n H(e) then e f m ðxÞ f n ðxÞ e for x 2 ½a; b: Theorem 7.1.5 implies that Z eðb aÞ a b Z fm b a f n eðb aÞ: Rb Since e > 0 is arbitrary, the sequence ð a f m Þ is a Cauchy sequence in R and therefore converges to some number, say A 2 R. We now show f 2 R½a; b with integral A. If e > 0 is given, let K(e) be such that if m > K(e), then j f m ðxÞ f ðxÞj < e for all x 2 ½a; b. If P_ :¼ fð½xi1 ; xi ; ti Þgni¼1 is any tagged partition of [a, b] and if m > K(e), then X n _ _ jSð f m ; PÞ Sð f ; PÞj ¼ f f m ðti Þ f ðti Þgðxi xi1 Þ i¼1 n X f m ðti Þ f ðti Þjðxi xi1 Þ i¼1 n X eðxi xi1 Þ ¼ eðb aÞ: i¼1 Rb We R b now choose r K(e) such that j a f r Aj < e and we let dr;e > 0 be such that _ < e whenever jjPjj _ < dr;e. Then we have j a f r Sð f r ; PÞj Z _ þ Sð f r ; PÞ _ _ Aj jSð f ; PÞ _ Sð f r ; PÞj jSð f ; PÞ eðb aÞ þ e þ e ¼ eðb a þ 2Þ: But since e > 0 is arbitrary, it follows that f 2 R½a; b and b a Rb a Z f r þ f ¼ A. b a f r A Q.E.D. The hypothesis of uniform convergence is a very stringent one and restricts the utility of this result. In Section 10.4 we will obtain some far-reaching generalizations of Theorem 8.2.4. For the present, we will state a result that does not require the uniformity of the convergence, but does require that the limit function be Riemann integrable. The proof is omitted. 8.2.5 Bounded Convergence Theorem Let ( fn) be a sequence in R½a; b that converges on [a, b] to a function f 2 R½a; b . Suppose also that there exists B > 0 such that j f n ðxÞj B for all x 2 ½a; b; n 2 N. Then equation (3) holds. C08 12/08/2010 14:48:19 252 Page 252 CHAPTER 8 SEQUENCES OF FUNCTIONS Dini’s Theorem We will end this section with a famous theorem due to Ulisse Dini (1845–1918) that gives a partial converse to Theorem 8.2.2 when the sequence is monotone. We will present a proof using nonconstant gauges (see Section 5.5). 8.2.6 Dini’s Theorem Suppose that (fn) is a monotone sequence of continuous functions on I :¼ [a, b] that converges on I to a continuous function f. Then the convergence of the sequence is uniform. Proof. We suppose that the sequence ( fn) is decreasing and let gm :¼ fm f. Then (gm) is a decreasing sequence of continuous functions converging on I to the 0-function. We will show that the convergence is uniform on I. Given e > 0, t 2 I, there exists me;t 2 N such that 0 gme;t ðtÞ < e=2. Since gme;t is continuous at t, there exists de ðtÞ > 0 such that 0 gme;t ðxÞ < e for all x 2 I satisfying jx 1j de ðtÞ. Thus, de is a gauge on I, and if P_ ¼ fðI i ; ti Þgni¼1 is a de -fine partition, we set M e :¼ maxfme;t1 ; . . . ; me;tn g. If m Me and x 2 I, then (by Lemma 5.5.3) there exists an index i with jx ti j de ðti Þ and hence 0 gm ðxÞ gm;ti ðxÞ < e: Therefore, the sequence (gm) converges uniformly to the 0-function. Q.E.D. It will be seen in the exercises that we cannot drop any one of the three hypotheses: (i) the functions fn are continuous, (ii) the limit function f is continuous, (iii) I is a closed bounded interval. Exercises for Section 8.2 1. Show that the sequence (xn=(1 þ xn)) does not converge uniformly on [0, 2] by showing that the limit function is not continuous on [0, 2]. 2. Prove that the sequence in Example 8.2.1(c) is an example of a sequence of continuous functions that converges nonuniformly to a continuous limit. 3. Construct a sequence of functions on [0, 1] each of which is discontinuous at every point of [0, 1] and which converges uniformly to a function that is continuous at every point. 4. Suppose ( fn) is a sequence of continuous functions on an interval I that converges uniformly on I to a function f. If (xn) I converges to x0 2 I, show that lim( fn(xn)) ¼ f (x0). 5. Let f : R ! R be uniformly continuous on R and let fn(x) :¼ f (x þ 1=n) for x 2 R . Show that ( fn) converges uniformly on R to f. 6. Let fn(x) :¼ 1=(1 þ x)n for x 2 [0, 1]. Find the pointwise limit f of the sequence ( fn) on [0, 1]. Does ( fn) converge uniformly to f on [0, 1]? 7. Suppose the sequence ( fn) converges uniformly to f on the set A, and suppose that each fn is bounded on A. (That is, for each n there is a constant Mn such that j f n ðxÞj M n for all x 2 A.) Show that the function f is bounded on A. 8. Let fn(x) :¼ nx=(1 þ nx2) for x 2 A :¼ [0, 1). Show that each fn is bounded on A, but the pointwise limit f of the sequence is not bounded on A. Does ( fn) converge uniformly to f on A? C08 12/08/2010 14:48:20 Page 253 8.3 THE EXPONENTIAL AND LOGARITHMIC FUNCTIONS 253 9. Let fn (x) :¼ xn=n for x 2 [0, 1]. Show that the sequence ( fn) of differentiable functions converges uniformly to a differentiable function f on [0, 1], and that the sequence ( f n0 ) converges on [0, 1] to a function g, but that gð1Þ 6¼ f 0 ð1Þ. 10. Let gn ðxÞ :¼ enx =n for x 0; n 2 N. Examine the relation between lim(gn) and lim( g0n ). 11. Let I :¼ [a, b] and let ( fn) be a sequence of functions on I ! R that converges on I to f. Suppose 0 that each derivative f n0 is continuous R x on I and that the 0sequence ( f n ) is uniformly convergent to g on I. Prove that f ðxÞ f ðaÞ ¼ a gðtÞdt and that f ðxÞ ¼ gðxÞ for all x 2 I. R2 2 12. Show that lim 1 enx dx ¼ 0. Rp 13. If a > 0, show that lim a ðsin nxÞ=ðnxÞdx ¼ 0. What happens if a ¼ 0? 14. Let f n ðxÞ :¼ nx=ð1 þ nxÞ for x 2 ½0; 1. Show that ( fn) converges nonuniformly to an integrable R1 R1 function f and that 0 f ðxÞdx ¼ lim 0 f n ðxÞdx. R1 15. Let gn ðxÞ :¼ nxð1 xÞn for x 2 ½0; 1; n 2 N. Discuss the convergence of (gn) and ð 0 gn dxÞ. 16. Let {r1, r2, . . . , rn . . . } be an enumeration of the rational numbers in I :¼ ½0; 1, and let f n : I ! R be defined to be 1 if x ¼ r1 ; . . . ; rn and equal to 0 otherwise. Show that fn is Riemann integrable for each n 2 N, that f 1 ðxÞ f 2 ðxÞ f n ðxÞ , and that f ðxÞ :¼ limð f n ðxÞÞ is the Dirichlet function, which is not Riemann integrable on [0, 1]. 17. Let f n ðxÞ :¼ 1 for x 2 ð0; 1=nÞ and f n ðxÞ :¼ 0 elsewhere in [0, 1]. Show that ( fn) is a decreasing sequence of discontinuous functions that converges to a continuous limit function, but the convergence is not uniform on [0, 1]. 18. Let f n ðxÞ :¼ xn for x 2 ½0; 1; n 2 N. Show that ( fn) is a decreasing sequence of continuous functions that converges to a function that is not continuous, but the convergence is not uniform on [0, 1]. 19. Let f n ðxÞ :¼ x=n for x 2 ½0; 1Þ; n 2 N. Show that ( fn) is a decreasing sequence of continuous functionsthatconvergestoa continuous limit function,butthe convergenceisnot uniformon [0, 1). 20. Give an example of a decreasing sequence ( fn) of continuous functions on [0, 1) that converges to a continuous limit function, but the convergence is not uniform on [0, 1). Section 8.3 The Exponential and Logarithmic Functions We will now introduce the exponential and logarithmic functions and will derive some of their most important properties. In earlier sections of this book we assumed some familiarity with these functions for the purpose of discussing examples. However, it is necessary at some point to place these important functions on a firm foundation in order to establish their existence and determine their basic properties. We will do that here. There are several alternative approaches one can take to accomplish this goal. We will proceed by first proving the existence of a function that has itself as derivative. From this basic result, we obtain the main properties of the exponential function. The logarithm function is then introduced as the inverse of the exponential function, and this inverse relation is used to derive the properties of the logarithm function. The Exponential Function We begin by establishing the key existence result for the exponential function. 8.3.1 Theorem There exists a function E : R ! R such that: (i) (ii) E0 ðxÞ ¼ EðxÞ f or all x 2 R. Eð0Þ ¼ 1. C08 12/08/2010 14:48:20 254 Page 254 CHAPTER 8 SEQUENCES OF FUNCTIONS Proof. We inductively define a sequence (En) of continuous functions as follows: ð1Þ E1 ðxÞ :¼ 1 þ x; Z ð2Þ Enþ1 ðxÞ :¼ 1 þ x 0 En ðtÞdt; for all n 2 N; x 2 R. Clearly E1 is continuous on R and hence is integrable over any bounded interval. If En has been defined and is continuous on R, then it is integrable over any bounded interval, so that Enþ1 is well-defined by the above formula. Moreover, it follows from the Fundamental Theorem (Second Form) 7.3.5 that Enþ1 is differentiable at any point x 2 R and that ð3Þ E0nþ1 ðxÞ ¼ En ðxÞ for n 2 N: An Induction argument (which we leave to the reader) shows that ð4Þ En ðxÞ ¼ 1 þ x x2 xn þ þ þ 1! 2! n! for x 2 R: Let A > 0 be given; then if jxj A and m > n > 2A, we have nþ1 x xm ð5Þ jEm ðxÞ En ðxÞj ¼ þ þ ðn þ 1Þ! m! " Anþ1 A A 1 þ þ þ n n ðn þ 1Þ! < mn1 # Anþ1 2: ðn þ 1Þ! Since lim(An=n!) ¼ 0, it follows that the sequence (En) converges uniformly on the interval [A, A] where A > 0 is arbitrary. In particular this means that (En(x)) converges for each x 2 R. We define E : R ! R by EðxÞ :¼ lim En ðxÞ for x 2 R: Since each x 2 R is contained inside some interval [A, A], it follows from Theorem 8.2.2 that E is continuous at x. Moreover, it is clear from (1) and (2) that En ð0Þ ¼ 1 for all n 2 N. Therefore Eð0Þ ¼ 1, which proves (ii). On any interval [A, A] we have the uniform convergence of the sequence (En). In view of (3), we also have the uniform convergence of the sequence (E0n ) of derivatives. It therefore follows from Theorem 8.2.3 that the limit function E is differentiable on [A, A] and that E0 ðxÞ ¼ limðE0n ðxÞÞ ¼ limðEn1 ðxÞÞ ¼ EðxÞ for all x 2 [A, A]. Since A > 0 is arbitrary, statement (i) is established. Q.E.D. 8.3.2 Corollary The function E has a derivative of every order and E (n)(x) ¼ E(x) for all n 2 N; x 2 R. C08 12/08/2010 14:48:20 Page 255 8.3 THE EXPONENTIAL AND LOGARITHMIC FUNCTIONS 255 Proof. If n ¼ 1, the statement is merely property (i). It follows for arbitrary n 2 N by Induction. Q.E.D. 8.3.3 Corollary If x > 0, then 1 þ x < E(x). Proof. It is clear from (4) that if x > 0, then the sequence (En (x)) is strictly increasing. Hence E1(x) < E(x) for all x > 0. Q.E.D. It is next shown that the function E, whose existence was established in Theorem 8.3.1, is unique. 8.3.4 Theorem The function E : R ! R that satisfies (i) and (ii) of Theorem 8.3.1 is unique. Proof. Let E1 and E2 be two functions on R to R that satisfy properties (i) and (ii) of Theorem 8.3.1 and let F :¼ E1 E2 . Then F 0 ðxÞ ¼ E01 ðxÞ E02 ðxÞ ¼ E1 ðxÞ E2 ðxÞ ¼ FðxÞ for all x 2 R and Fð0Þ ¼ E1 ð0Þ E2 ð0Þ ¼ 1 1 ¼ 0: It is clear (by Induction) that F has derivatives of all orders and indeed that F ðnÞ ðxÞ ¼ FðxÞ for n 2 N; x 2 R. Let x 2 R be arbitrary, and let Ix be the closed interval with endpoints 0, x. Since F is continuous on Ix, there exists K > 0 such that jFðtÞj K for all t 2 I x . If we apply Taylor’s Theorem 6.4.1 to F on the interval Ix and use the fact that F (k)(0) ¼ F(0) ¼ 0 for all k 2 N, it follows that for each n 2 N there is a point cn 2 Ix such that FðxÞ ¼ Fð0Þ þ ¼ F 0 ð0Þ F ðn1Þ n1 F ðnÞ ðcn Þ n x þ þ x x þ 1! n! ðn 1Þ! Fðcn Þ n x : n! Therefore we have jFðxÞj Kjxjn n! for all n 2 N: But since lim(jxj=n!) ¼ 0, we conclude that F(x) ¼ 0. Since x 2 R is arbitrary, we infer that E1 ðxÞ E2 ðxÞ ¼ FðxÞ ¼ 0 for all x 2 R. Q.E.D. The standard terminology and notation for the function E (which we now know exists and is unique) is given in the following definition. 8.3.5 Definition The unique function E : R ! R, such that E0 ðxÞ ¼ EðxÞ for all x 2 R and E(0) ¼ 1, is called the exponential function. The number e :¼ E(1) is called Euler’s number. We will frequently write expðxÞ :¼ EðxÞ or ex :¼ EðxÞ for x 2 R: The number e can be obtained as a limit, and thereby approximated, in several different ways. [See Exercises 1 and 10, and Example 3.3.6.] The use of the notation ex for E(x) is justified by property (v) in the next theorem, where it is noted that if r is a rational number, then E(r) and er coincide. (Rational C08 12/08/2010 14:48:21 256 Page 256 CHAPTER 8 SEQUENCES OF FUNCTIONS exponents were discussed in Section 5.6.) Thus, the function E can be viewed as extending the idea of exponentiation from rational numbers to arbitrary real numbers. For a definition of ax for a > 0 and arbitrary x 2 R, see Definition 8.3.10. 8.3.6 Theorem The exponential function satisfies the following properties: (iii) EðxÞ 6¼ 0 f or all x 2 R; (iv) Eðx þ yÞ ¼ EðxÞEðyÞ f or all x; y 2 R; (v) EðrÞ ¼ er f or all r 2 Q : Proof. (iii) Let a 2 R be such that E(a) ¼ 0, and let Ja be the closed interval with endpoints 0, a. Let K jEðtÞj for all t 2 J a . Taylor’s Theorem 6.4.1 implies that for each n 2 N there exists a point cn 2 J a such that 1 ¼ Eð0Þ ¼ EðaÞ þ þ E0 ðaÞ Eðn1Þ ðaÞ ðaÞ þ þ ðaÞn1 1! ðn 1Þ! EðnÞ ðaÞ Eðcn Þ ðaÞn ¼ ðaÞn: ðnÞ! n! Thus we have 0 < 1 ðK=n!Þjajn for n 2 N. But since limðjajn =n!Þ ¼ 0, this is a contradiction. (iv) Let y be fixed; by (iii) we have EðyÞ 6¼ 0. Let G : R ! R be defined by GðxÞ :¼ Eðx þ yÞ EðyÞ for x 2 R: Evidently we have G0 ðxÞ ¼ E0 ðx þ yÞ=EðyÞ ¼ Eðx þ yÞ=EðyÞ ¼ GðxÞ for all x 2 R, and Gð0Þ ¼ Eð0 þ yÞ=EðyÞ ¼ 1. It follows from the uniqueness of E, proved in Theorem 8.3.4, that GðxÞ ¼ EðxÞ for all x 2 R. Hence Eðx þ yÞ ¼ EðxÞEðyÞ for all x 2 R. Since y 2 R is arbitrary, we obtain (iv). (v) It follows from (iv) and Induction that if n 2 N; x 2 R, then EðnxÞ ¼ EðxÞn : If we let x ¼ 1=n, this relation implies that e ¼ Eð1Þ ¼ E n 1 n ¼ E 1 n n ; whence it follows that Eð1=nÞ ¼ e1=n . Also we have EðmÞ ¼ 1=EðmÞ ¼ 1=em ¼ em for m 2 N. Therefore, if m 2 Z, n 2 N, we have Eðm=nÞ ¼ ðEð1=nÞÞm ¼ ðe1=n Þm ¼ em=n : This establishes (v). Q.E.D. 8.3.7 Theorem The exponential function E is strictly increasing on R and has range equal to fy 2 R : y > 0g. Further, we have (vi) lim EðxÞ ¼ 0 and x!1 lim EðxÞ ¼ 1: x!1 Proof. We know that Eð0Þ ¼ 1 > 0 and EðxÞ 6¼ 0 for all x 2 R. Since E is continuous on R, it follows from Bolzano’s Intermediate Value Theorem 5.3.7 that EðxÞ > 0 for all x 2 R. Therefore E0 ðxÞ ¼ EðxÞ > 0 for x 2 R, so that E is strictly increasing on R. C08 12/08/2010 14:48:21 Page 257 8.3 THE EXPONENTIAL AND LOGARITHMIC FUNCTIONS 257 It follows from Corollary 8.3.3 that 2 < e and that lim EðxÞ ¼ 1. Also, if z > 0, then x!1 since 0 < EðzÞ ¼ 1=EðzÞ it follows that lim EðxÞ ¼ 0. Therefore, by the Intermediate x!1 Value Theorem 5.3.7, every y 2 R with y > 0 belongs to the range of E. Q.E.D. The Logarithm Function We have seen that the exponential function E is a strictly increasing differentiable function with domain R and range fy 2 R : y > 0g. (See Figure 8.3.1.) It follows that R has an inverse function. Figure 8.3.1 Graph of E Figure 8.3.2 Graph of L 8.3.8 Definition The function inverse to E : R ! R is called the logarithm (or the natural logarithm). (See Figure 8.3.2.) It will be denoted by L , or by ln. Since E and L are inverse functions, we have ðL EÞðxÞ ¼ x for all x 2 R and ðE LÞðyÞ ¼ y for all y 2 R; y > 0: These formulas may also be written in the form ln ex ¼ x; eln y ¼ y: 8.3.9 Theorem The logarithm is a strictly increasing function L with domain fx 2 R : x > 0g and range R. The derivative of L is given by (vii) L0 ðxÞ ¼ 1=x f or x > 0: The logarithm satisfies the functional equation (viii) LðxyÞ ¼ LðxÞ þ LðyÞ for x > 0; y > 0: Moreover, we have (ix) LðlÞ ¼ 0 and LðeÞ ¼ 1; (x) Lðxr Þ ¼ rLðxÞ for x > 0; r 2 Q : (xi) lim LðxÞ ¼ 1 and lim LðxÞ ¼ 1: x!0þ x!1 C08 12/08/2010 14:48:22 258 Page 258 CHAPTER 8 SEQUENCES OF FUNCTIONS Proof. That L is strictly increasing with domain fx 2 R : x > 0g and range R follows from the fact that E is strictly increasing with domain R and range fy 2 R : y > 0g. (vii) Since E0 ðxÞ ¼ EðxÞ > 0, it follows from Theorem 6.1.9 that L is differentiable on (0, 1) and that 1 1 1 ¼ for x 2 ð0; 1Þ: ¼ L0 ðxÞ ¼ 0 ðE LÞðxÞ ðE LÞðxÞ x (viii) If x > 0, y > 0, let u :¼ L(x) and v :¼ L(y). Then we have x ¼ E(u) and y ¼ E(v). It follows from property (iv) of Theorem 8.3.6 that xy ¼ EðuÞEðvÞ ¼ Eðu þ vÞ; so that LðxyÞ ¼ ðL EÞðu þ vÞ ¼ u þ v ¼ LðxÞ þ LðyÞ. This establishes (viii). The properties in (ix) follow from the relations Eð0Þ ¼ 1 and Eð1Þ ¼ e. (x) This result follows from (viii) and Mathematical Induction for n 2 N, and is extended to r 2 Q by arguments similar to those in the proof of 8.3.6(v). To establish property (xi), we first note that since 2 < e, then lim(en) ¼ 1 and lim(en) ¼ 0. Since L(en) ¼ n and L(en) ¼ n it follows from the fact that L is strictly increasing that lim LðxÞ ¼ lim Lðen Þ ¼ 1 x!1 and lim LðxÞ ¼ lim Lðen Þ ¼ 1: x!0þ Q.E.D. Power Functions In Definition 5.6.6, we discussed the power function x 7! xr ; x > 0, where r is a rational number. By using the exponential and logarithm functions, we can extend the notion of power functions from rational to arbitrary real powers. 8.3.10 Definition If a 2 R and x > 0, the number xa is defined to be xa :¼ ea ln x ¼ EðaLðxÞÞ: The function x 7! xa for x > 0 is called the power function with exponent a. Note If x > 0 and a ¼ m=n where m 2 Z; n 2 N, then we defined xa :¼ ðxm Þ1=n in a Section 5.6. Hence we have ln xa ¼ a ln x, whence xa ¼ eln x ¼ ea ln x . Hence Definition 8.3.10 is consistent with the definition given in Section 5.6. We now state some properties of the power functions. Their proofs are immediate consequences of the properties of the exponential and logarithm functions and will be left to the reader. 8.3.11 Theorem If a 2 R and x, y belong to (0, 1), then: ( a ) 1a ¼ 1; ( c ) ðxyÞa ¼ xa ya; ( b ) xa > 0; ( d ) ðx=yÞa ¼ xa =ya: 8.3.12 Theorem If a; b 2 R and x 2 ð0; 1Þ, then: (a) (c) xaþb ¼ xa xb xa ¼ 1=xa; (b) (d) ðxa Þb ¼ xab ¼ ðxb Þa; if a < b; then xa < xb f or x > 1: The next result concerns the differentiability of the power functions. C08 12/08/2010 14:48:22 Page 259 8.3 THE EXPONENTIAL AND LOGARITHMIC FUNCTIONS 259 8.3.13 Theorem Let a 2 R . Then the function x 7! xa on (0, 1) to R is continuous and differentiable, and Dxa ¼ axa1 for x 2 ð0; 1Þ: Proof. By the Chain Rule we have Dxa ¼ Dea ln x ¼ ea ln x Dða ln xÞ a1 ¼ xa a for x 2 ð0; 1Þ: x ¼ ax Q.E.D. It will be seen in an exercise that if a > 0, the power function x 7! xa is strictly increasing on (0, 1) to R, and that if a < 0, the function x 7! xa is strictly decreasing. (What happens if a ¼ 0?) The graphs of the functions x 7! xa on (0, 1) to R are similar to those in Figure 5.6.8. The Function loga If a > 0, a 6¼ 1, it is sometimes useful to define the function loga. 8.3.14 Definition Let a > 0, a 6¼ 1. We define loga ðxÞ :¼ ln x ln a for x 2 ð0; 1Þ: For x 2 (0, 1), the number loga(x) is called the logarithm of x to the base a. The case a ¼ e yields the logarithm (or natural logarithm) function of Definition 8.3.8. The case a ¼ 10 gives the base 10 logarithm (or common logarithm) function log10 often used in computations. Properties of the functions loga will be given in the exercises. Exercises for Section 8.3 1. Show that if x > 0 and if n > 2x, then n x e 1 þ x þ þ x 1! n! nþ1 < 2x ðn þ 1Þ! Use this formula to show that 2 23 < e < 2 34, hence e is not an integer. 2. Calculate e correct to five decimal places. 3. Show that if 0 x a and n 2 N, then 1þ x xn x xn1 ea xn þ : þ þ ex 1 þ þ þ n! ðn 1Þ! n! 1! 1! 4. Show that if n 2, then 1 1 e þ þ n! < < 1: 2! n! nþ1 Use this inequality to prove that e is not a rational number. 0 < en! 1 þ 1 þ 5. If x 0 and n 2 N, show that 1 ðxÞn : ¼ 1 x þ x2 x3 þ þ ðxÞn1 þ 1þx xþ1 Use this to show that lnðx þ 1Þ ¼ x x2 x3 xn þ þ ð1Þn1 þ 2 3 n Z 0 x ðtÞn dt 1þt C08 12/08/2010 14:48:22 260 Page 260 CHAPTER 8 SEQUENCES OF FUNCTIONS and that 2 3 n lnðx þ 1Þ x x þ x þ ð1Þn1 x 2 3 n nþ1 x n þ 1: 6. Use the formula in the preceding exercise to calculate ln 1.1 and ln 1.4 accurate to four decimal places. How large must one choose n in this inequality to calculate ln 2 accurate to four decimal places? 7. Show that ln(e=2) ¼ 1 ln 2. Use this result to calculate ln 2 accurate to four decimal places. 8. Let f : R ! R be such that f 0(x) ¼ f (x) for all x 2 R. Show that there exists K 2 R such that f (x) ¼ Kex for all x 2 R. 9. Let ak > 0 for k ¼ 1, . . . , n and let A :¼ ða1 þ þ an Þ=n be the arithmetic mean of these numbers. For each k, put xk :¼ ak =A 1 in the inequality 1 þ x ex. Multiply the resulting terms to prove the Arithmetic–Geometric Mean Inequality 1 ða1 an Þ1=n ða1 þ þ an Þ: n Moreover, show that equality holds in (6) if and only if a1 ¼ a2 ¼ ¼ an . 10. Evaluate L0(1) by using the sequence (1 þ 1=n) and the fact that e ¼ lim ð1 þ 1=nÞn . ð6Þ 11. Establish the assertions in Theorem 8.3.11. 12. Establish the assertions in Theorem 8.3.12. 13. (a) Show that if a > 0, then the function x 7! xa is strictly increasing on (0, 1) to R and that lim xa ¼ 0 and lim xa ¼ 1. x!0þ x!1 (b) Show that if a < 0, then the function x 7! xa is strictly decreasing on (0, 1) to R and that lim xa ¼ 1 and lim xa ¼ 0. x!0þ x!1 14. Prove that if a > 0, a 6¼ 1, then aloga x ¼ x for all x 2 (0, 1) and loga(ay) ¼ y for all y 2 R. Therefore the function x 7! loga x on (0, 1) to R is inverse to the function y 7! ay on R. 15. If a > 0, a 6¼ 1, show that the function x 7! loga x is differentiable on (0, 1) and that D logax ¼ 1=(x ln a) for x 2 (0, 1). 16. If a > 0, a 6¼ 1, and x and y belong to (0, 1), prove that loga (xy) ¼ logax þ logay. 17. If a > 0, a 6¼ 1, and b > 0, b 6¼ 1, show that loga x ¼ ln b logb x ln a for x 2 ð0; 1Þ: In particular, show that log10x ¼ (ln e=ln 10) ln x ¼ (log10e) ln x for x 2 (0, 1). Section 8.4 The Trigonometric Functions Along with the exponential and logarithmic functions, there is another very important collection of transcendental functions known as the ‘‘trigonometric functions.’’ These are the sine, cosine, tangent, cotangent, secant, and cosecant functions. In elementary courses, they are usually introduced on a geometric basis in terms of either triangles or the unit circle. In this section, we introduce the trigonometric functions in an analytical manner and then establish some of their basic properties. In particular, the various properties of the trigonometric functions that were used in examples in earlier parts of this book will be derived rigorously in this section. It suffices to deal with the sine and cosine since the other four trigonometric functions are defined in terms of these two. Our approach to the sine and cosine is similar in spirit to C08 12/08/2010 14:48:23 Page 261 8.4 THE TRIGONOMETRIC FUNCTIONS 261 our approach to the exponential function in that we first establish the existence of functions that satisfy certain differentiation properties. 8.4.1 Theorem There exist functions C : R ! R and S : R ! R such that (i) (ii) C 00 ðxÞ ¼ CðxÞ and S00 ðxÞ ¼ SðxÞ f or all x 2 R: Cð0Þ ¼ 1; C0 ð0Þ ¼ 0; and Sð0Þ ¼ 0; S0 ð0Þ ¼ 1: Proof. We define the sequences (Cn) and (Sn) of continuous functions inductively as follows: ð1Þ C1 ðxÞ :¼ 1; S1 ðxÞ :¼ x; Z x Sn ðxÞ :¼ Cn ðtÞdt; ð2Þ 0 Cnþ1 ðxÞ :¼ 1 ð3Þ Z 0 x Sn ðtÞdt; for all n 2 N, x 2 R. One sees by Induction that the functions Cn and Sn are continuous on R and hence they are integrable over any bounded interval; thus these functions are well-defined by the above formulas. Moreover, it follows from the Fundamental Theorem 7.3.5 that Sn and Cnþ1 are differentiable at every point and that ð4Þ Sn0 ðxÞ ¼ C n ðxÞ and 0 Cnþ1 ðxÞ ¼ Sn ðxÞ for n 2 N; x 2 R: Induction arguments (which we leave to the reader) show that C nþ1 ðxÞ ¼ 1 x2 x4 x2n þ þ ð1Þn ; 2! 4! ð2nÞ! Snþ1 ðxÞ ¼ x x3 x5 x2nþ1 þ þ ð1Þn : 3! 5! ð2n þ 1Þ! Let A > 0 be given. Then if jxj A and m > n > 2A, we have that (since A=2n < 1=4): 2n x x2nþ2 x2m2 C þ ðxÞ C ðxÞ ¼ ð5Þ j m j n ð2nÞ! ð2n þ 2Þ! ð2m 2Þ! " # A2n A 2 A 2m2n2 1þ þ þ 2n 2n ð2nÞ! < A2n 16 : ð2nÞ! 15 Since limðA2n =ð2nÞ!Þ ¼ 0, the sequence (Cn) converges uniformly on the interval ½A; A, where A > 0 is arbitrary. In particular, this means that (C n ðxÞ) converges for each x 2 R. We define C : R ! R by for x 2 R: CðxÞ :¼ lim Cn ðxÞ It follows from Theorem 8.2.2 that C is continuous on R and, since Cn ð0Þ ¼ 1 for all n 2 N, that Cð0Þ ¼ 1. If jxj A and m n > 2A, it follows from (2) that Z x Sm ðxÞ Sn ðxÞ ¼ fCm ðtÞ C n ðtÞgdt: 0 C08 12/08/2010 14:48:23 262 Page 262 CHAPTER 8 SEQUENCES OF FUNCTIONS If we use (5) and Corollary 7.3.15, we conclude that jSm ðxÞ Sn ðxÞj A2n 16 A ; ð2nÞ! 15 whence the sequence (Sn) converges uniformly on ½A; A. We define S : R ! R by SðxÞ :¼ lim Sn ðxÞ for x 2 R: It follows from Theorem 8.2.2 that S is continuous on R and, since Sn ð0Þ ¼ 0 for all n 2 N, that Sð0Þ ¼ 0. Since Cn0 ðxÞ ¼ Sn1 ðxÞ for n > 1, it follows from the above that the sequence 0 (Cn ) converges uniformly on ½A; A. Hence by Theorem 8.2.3, the limit function C is differentiable on ½A; A and C0 ðxÞ ¼ lim C 0n ðxÞ ¼ limðSn1 ðxÞÞ ¼ SðxÞ for x 2 ½A; A: Since A > 0 is arbitrary, we have ð6Þ C 0 ðxÞ ¼ SðxÞ A similar argument, based on the fact that R and that ð7Þ Sn0 ðxÞ S0 ðxÞ ¼ CðxÞ for x 2 R: ¼ C n ðxÞ, shows that S is differentiable on for all x 2 R: It follows from (6) and (7) that C 00 ðxÞ ¼ ðSðxÞÞ0 ¼ CðxÞ and S00 ðxÞ ¼ ðCðxÞÞ0 ¼ SðxÞ for all x 2 R. Moreover, we have C 0 ð0Þ ¼ Sð0Þ ¼ 0; S0 ð0Þ ¼ Cð0Þ ¼ 1: Thus statements (i) and (ii) are proved. Q.E.D. 8.4.2 Corollary If C, S are the functions in Theorem 8.4.1, then (iii) C 0 ðxÞ ¼ SðxÞ and S0 ðxÞ ¼ CðxÞ for x 2 R. Moreover, these functions have derivatives of all orders. Proof. The formulas (iii) were established in (6) and (7). The existence of the higher order derivatives follows by Induction. Q.E.D. 8.4.3 Corollary The functions C and S satisfy the Pythagorean Identity: (iv) ðCðxÞÞ2 þ ðSðxÞÞ2 ¼ 1 for x 2 R. Proof. Let f ðxÞ :¼ ðCðxÞÞ2 þ ðSðxÞÞ2 for x 2 R, so that f 0 ðxÞ ¼ 2CðxÞðSðxÞÞ þ 2SðxÞðCðxÞÞ ¼ 0 for x 2 R: Thus it follows that f (x) is a constant for all x 2 R. But since f ð0Þ ¼ 1 þ 0 ¼ 1, we conclude that f ðxÞ ¼ 1 for all x 2 R. Q.E.D. We next establish the uniqueness of the functions C and S. 8.4.4 Theorem The functions C and S satisfying properties (i) and (ii) of Theorem 8.4.1 are unique. C08 12/08/2010 14:48:24 Page 263 8.4 THE TRIGONOMETRIC FUNCTIONS 263 Proof. Let C1 and C2 be two functions on R to R that satisfy C 00j ðxÞ ¼ Cj ðxÞ for all x 2 R and Cj ð0Þ ¼ 1, Cj0ð0Þ ¼ 0 for j ¼ 1; 2. If we let D :¼ C1 C2 , then D00 ðxÞ ¼ DðxÞ for x 2 R and Dð0Þ ¼ 0 and DðkÞ ð0Þ ¼ 0 for all k 2 N. Now let x 2 R be arbitrary, and let I x be the interval with endpoints 0; x. Since D ¼ C 1 C2 and T :¼ S1 S2 ¼ C20 C10 are continuous on I x , there exists K > 0 such that jDðtÞj K and jTðtÞj K for all t 2 I x . If we apply Taylor’s Theorem 6.4.1 to D on I x and use the fact that Dð0Þ ¼ 0; DðkÞ ð0Þ ¼ 0 for k 2 N, it follows that for each n 2 N there is a point cn 2 I x such that DðxÞ ¼ Dð0Þ þ ¼ D0 ð0Þ Dðn1Þ ð0Þ n1 DðnÞ ðcn Þ n x þ þ x x þ 1! ðn 1Þ! n! DðnÞ ðcn Þ n x : n! Now either DðnÞ ðcn Þ ¼ Dðcn Þ or DðnÞ ðcn Þ ¼ jDðxÞj Tðcn Þ. In either case we have K jxjn : n! But since lim ðjxjn =n!Þ ¼ 0, we conclude that DðxÞ ¼ 0. Since x 2 R is arbitrary, we infer that C1 ðxÞ C 2 ðxÞ ¼ 0 for all x 2 R. A similar argument shows that if S1 and S2 are two functions on R ! R such that S00j ðxÞ ¼ Sj ðxÞ for all x 2 R and Sj ð0Þ ¼ 0, S0j ð0Þ ¼ 1 for j ¼ 1, 2, then we have S1 ðxÞ ¼ S2 ðxÞ for all x 2 R. Q.E.D. Now that existence and uniqueness of the functions C and S have been established, we shall give these functions their familiar names. 8.4.5 Definition The unique functions C : R ! R and S : R ! R such that C00 ðxÞ ¼ CðxÞ and S00 ðxÞ ¼ SðxÞ for all x 2 R and Cð0Þ ¼ 1, C0 ð0Þ ¼ 0, and Sð0Þ ¼ 0, S0 ð0Þ ¼ 1, are called the cosine function and the sine function, respectively. We ordinarily write cos x :¼ CðxÞ and sin x :¼ SðxÞ for x 2 R: The differentiation properties in (i) of Theorem 8.4.1 do not by themselves lead to uniquely determined functions. We have the following relationship. 8.4.6 Theorem If f : R ! R is such that f 00 ðxÞ ¼ f ðxÞ for x 2 R; then there exist real numbers a, b such that f ðxÞ ¼ aCðxÞ þ bSðxÞ for x 2 R: Proof. Let gðxÞ :¼ f ð0ÞCðxÞ þ f 0 ð0ÞSðxÞ for x 2 R. It is readily seen that g00 ðxÞ ¼ gðxÞ and that gð0Þ ¼ f ð0Þ, and since g0 ðxÞ ¼ f ð0ÞSðxÞ þ f 0 ð0ÞCðxÞ; that g0 ð0Þ ¼ f 0 ð0Þ. Therefore the function h :¼ f g is such that h00 ðxÞ ¼ hðxÞ for all x 2 R and hð0Þ ¼ 0, h0 ð0Þ ¼ 0. Thus it follows from the proof of the preceding theorem that hðxÞ ¼ 0 for all x 2 R. Therefore f ðxÞ ¼ gðxÞ for all x 2 R. Q.E.D. C08 12/08/2010 14:48:24 264 Page 264 CHAPTER 8 SEQUENCES OF FUNCTIONS We shall now derive a few of the basic properties of the cosine and sine functions. 8.4.7 Theorem The function C is even and S is odd in the sense that (v) CðxÞ ¼ CðxÞ and SðxÞ ¼ SðxÞ for x 2 R. If x, y 2 R, then we have the ‘‘addition formulas’’ (vi) Cðx þ yÞ ¼ CðxÞCðyÞ SðxÞSðyÞ; Sðx þ yÞ ¼ SðxÞCðyÞ þ CðxÞSðyÞ. Proof. (v) If wðxÞ :¼ CðxÞ for x 2 R, then a calculation shows that w00 ðxÞ ¼ wðxÞ for x 2 R. Moreover, wð0Þ ¼ 1 and w0 ð0Þ ¼ 0 so that w ¼ C. Hence, CðxÞ ¼ CðxÞ for all x 2 R. In a similar way one shows that SðxÞ ¼ SðxÞ for all x 2 R. (vi) Let y 2 R be given and let f ðxÞ :¼ Cðx þ yÞ for x 2 R. A calculation shows that f 00 ðxÞ ¼ f ðxÞ for x 2 R. Hence, by Theorem 8.4.6, there exists real numbers a, b such that f ðxÞ ¼ Cðx þ yÞ ¼ aCðxÞ þ bSðxÞ and f 0 ðxÞ ¼ Sðx þ yÞ ¼ aSðxÞ þ bCðxÞ for x 2 R. If we let x ¼ 0, we obtain CðyÞ ¼ a and SðyÞ ¼ b, whence the first formula in (vi) follows. The second formula is proved similarly. Q.E.D. The following inequalities were used earlier (for example, in 4.2.8). 8.4.8 Theorem If x 2 R, x 0, then we have (vii) x SðxÞ x; (viii) 1 12 x2 CðxÞ 1; (ix) x 16 x3 SðxÞ x; (x) 1 4 1 12 x2 CðxÞ 1 12 x2 þ 24 x . Proof. Corollary 8.4.3 implies that 1 CðtÞ 1 for t 2 R, so that if x 0, then Z x x CðtÞdt x; 0 whence we have (vii). If we integrate (vii), we obtain Z x 12 x2 SðtÞ dt 12 x2 ; 0 whence we have 12 x2 CðxÞ þ 1 12 x2 : Thus we have 1 12 x2 CðxÞ, which implies (viii). Inequality (ix) follows by integrating (viii), and (x) follows by integrating (ix). Q.E.D. The number p is obtained via the following lemma. pffiffiffi pffiffiffi 8.4.9 Lemma There exists a root g of the cosine function in the interval ( 2; 3). Moreover CðxÞ > 0 for x 2 ½0; gÞ. The number 2g is the smallest positive root of S. Proof. Inequality (x) of Theorem 8.4.8 implies that C has a root between the positive root pffiffiffi 2 of x2 2 ¼ 0 and the smallest positive root of x4 12x2 þ 24 ¼ 0, which is pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi 6 2 3 < 3. We let g be the smallest such root of C. C08 12/08/2010 14:48:25 Page 265 8.4 THE TRIGONOMETRIC FUNCTIONS 265 It follows from the second formula in (vi) with x ¼ y that Sð2xÞ ¼ 2SðxÞCðxÞ. This relation implies that Sð2gÞ ¼ 0, so that 2g is a positive root of S. The same relation implies that if 2d > 0 is the smallest positive root of S, then CðdÞ ¼ 0. Since g is the smallest positive root of C, we have d ¼ g. Q.E.D. 8.4.10 Definition Let p :¼ 2g denote the smallest positive root of S. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi Note The inequality 2 < g < 6 2 3 implies that 2:828 < p < 3:185. 8.4.11 Theorem The functions C and S have period 2p in the sense that (xi) Cðx þ 2pÞ ¼ CðxÞ and Sðx þ 2pÞ ¼ SðxÞ for x 2 R. Moreover we have (xii) SðxÞ ¼ C 12p x ¼ C x þ 12 p ; x 2 R. CðxÞ ¼ S 12 p x ¼ S x þ 12 p for all Proof. (xi) Since Sð2xÞ ¼ 2SðxÞCðxÞ and SðpÞ ¼ 0, then Sð2pÞ ¼ 0. Further, if x ¼ y in (vi), we obtain Cð2xÞ ¼ ðCðxÞÞ2 ðSðxÞÞ2 . Therefore Cð2pÞ ¼ 1. Hence (vi) with y ¼ 2p gives Cðx þ 2pÞ ¼ CðxÞCð2pÞ SðxÞSð2pÞ ¼ CðxÞ; and Sðx þ 2pÞ ¼ SðxÞCð2pÞ þ CðxÞSð2pÞ ¼ SðxÞ: (xii) We note that C 12 p ¼ 0, and it is an exercise to show that S 12 p ¼ 1. If we employ these together with formulas (vi), the desired relations are obtained. Q.E.D. Exercises for Section 8.4 1. Calculate cos(.2), sin(.2) and cos 1, sin 1 correct to four decimal places. 2. Show that jsin xj 1 and jcos xj 1 for all x 2 R. 3. Show that property (vii) of Theorem 8.4.8 does not hold if x < 0, but that we have jsin xj jxj for all x 2 R. Also show that jsin x xj jxj3 =6 for all x 2 R. 4. Show that if x > 0 then 1 x2 x4 x6 x2 x4 þ cos x 1 þ : 2 24 720 2 24 Use this inequality to establish a lower bound for p. 5. Calculate p by approximating the smallest positive zero of sin. (Either bisect intervals or use Newton’s Method of Section 6.4.) 6. Define the sequence (cn) and (sn) inductively by c1 ðxÞ :¼ 1; s1 ðxÞ :¼ x, and Z x Z x sn ðxÞ :¼ cn ðtÞdt; cnþ1 ðxÞ :¼ 1 þ sn ðtÞdt 0 0 for all n 2 N, x 2 R. Reason as in the proof of Theorem 8.4.1 to conclude that there exist functions c : R ! R and s : R ! R such that (j) c00 ðxÞ ¼ cðxÞ and s00 ðxÞ ¼ sðxÞ for all x 2 R, and (jj) cð0Þ ¼ 1, c0 ð0Þ ¼ 0 and sð0Þ ¼ 0, s0 ð0Þ ¼ 1. Moreover, c0 ðxÞ ¼ sðxÞ and s0 ðxÞ ¼ cðxÞ for all x 2 R. C08 12/08/2010 14:48:25 266 Page 266 CHAPTER 8 SEQUENCES OF FUNCTIONS 7. Show that the functions c, s in the preceding exercise have derivatives of all orders, and that they satisfy the identity ðcðxÞÞ2 ðsðxÞÞ2 ¼ 1 for all x 2 R. Moreover, they are the unique functions satisfying (j) and (jj). (The functions c, s are called the hyperbolic cosine and hyperbolic sine functions, respectively.) 8. If f : R ! R is such that f 00 ðxÞ ¼ f ðxÞ for all x 2 R, show that there exist real numbers a, b such that f ðxÞ ¼ acðxÞ þ bsðxÞ for all x 2 R. Apply this to the functions f 1 ðxÞ :¼ ex and f 2 ðxÞ :¼ ex for x 2 R. Show that cðxÞ ¼ 12 ðex þ ex Þ and sðxÞ ¼ 12 ðex ex Þ for x 2 R. 9. Show that the functions c, s in the preceding exercises are even and odd, respectively, and that cðx þ yÞ ¼ cðxÞcðyÞ þ sðxÞsðyÞ; sðx þ yÞ ¼ sðxÞcðyÞ þ cðxÞsðyÞ; for all x; y 2 R. 10. Show that cðxÞ 1 for all x 2 R, that both c and s are strictly increasing on (0; 1), and that lim cðxÞ ¼ lim sðxÞ ¼ 1. x!1 x!1 C09 12/08/2010 15:9:16 Page 267 CHAPTER 9 INFINITE SERIES In Section 3.7 we gave a brief introduction to the theory of infinite series. The reader will do well to look over that section at this time, since we will not repeat the definitions and results given there. Instead, in Section 9.1 we will introduce the important notion of the ‘‘absolute convergence’’ of a series. In Section 9.2 we will present some ‘‘tests’’ for absolute convergence that will probably be familiar to the reader from calculus. The third section gives a discussion of series that are not absolutely convergent. In the final section we study series of functions and will establish the basic properties of power series, which are very important in applications. Section 9.1 Absolute Convergence We have already met (in Section 3.7) a number of infinite series that are convergent and others that are divergent. For example, in Example 3.7.6(b) we saw that the harmonic series: 1 X 1 n¼1 n is divergent since its sequence of partial sums sn :¼ 11 þ 12 þ þ 1n ðn 2 N Þ is unbounded. On the other hand, we saw in Example 3.7.6(f) that the alternating harmonic series: 1 X ð1Þnþ1 n¼1 n is convergent because of the subtraction that takes place. Since ð1Þnþ1 1 ¼ ; n n P P these two series illustrate the fact that a series xn may be convergent, but the series jxn j obtained by taking the absolute values of the terms may be divergent. This observation leads us to an important definition. P 9.1.1 Definition Let XP :¼ (xn) be a sequence in R. We say that the series xn is absolutely convergent if the series jxn j is convergent in R. A series is said to be conditionally (or nonabsolutely) convergent if it is convergent, but it is not absolutely convergent. It is trivial that a series of positive terms is absolutely convergent if and only if it is convergent. We have noted above that the alternating harmonic series is conditionally convergent. 267 C09 12/08/2010 15:9:16 268 Page 268 CHAPTER 9 INFINITE SERIES 9.1.2 Theorem If a series in R is absolutely convergent, then it is convergent. P Proof. Since jxn j is convergent, the Cauchy Criterion 3.7.4 implies that, given e > 0 there exists M(e) 2 N such that if m > n M(e), then jxnþ1 j þ jxnþ1 j þ þ jxm j < e: However, by the Triangle Inequality, the left side of this expression dominates: jsm sn j ¼ jxnþ1 þ xnþ2 þ þ xm j: P Since e > 0 is arbitrary, Cauchy’s Criterion implies that xn converges. Q.E.D. Grouping of Series P P Given a series xn , we can construct many other series yk by leaving the order of the terms xn fixed, but inserting parentheses that group together finite numbers of terms. For example, the series indicated by 1 1 1 1 1 1 1 1 1 þ þ 1 þ þ þ 2 3 4 5 6 7 8 9 13 is obtained by grouping the terms in the alternating harmonic series. It is an interesting fact that such grouping does not affect the convergence or the value of a convergent series. P 9.1.3 Theorem If a series xn is convergent, then any series obtained from it by grouping the terms is also convergent and to the same value. Proof. Suppose that we have y1 :¼ x1 þ þ xk1 ; y2 :¼ xk1 þ1 þ þ xk2 ; : P P If sn denotes the nth partial sum of xn and tk denotes the kth partial sum of yk , then we have t 1 ¼ y 1 ¼ sk 1 ; t 2 ¼ y 1 þ y 2 ¼ sk 2 ; : P Thus, the sequence (tk) of partial Psums of the grouped series yk is a subsequence of the sequence (sn) of partial sums P of xn . Since this latter series was assumed to be convergent, so is the grouped series yk . Q.E.D. It is clear that the converse to this theorem is not true. Indeed, the grouping ð1 1Þ þ ð1 1Þ þ ð1 1Þ þ produces a convergent series from 1 X ð1Þn , which was seen to be divergent in Example n¼0 3.7.2(b) since the terms do not approach 0. Rearrangements of Series Loosely speaking, a ‘‘rearrangement’’ of a series is another series that is obtained from the given one by using all of the terms exactly once, but scrambling the order in which the terms are taken. For example, the harmonic series has rearrangements 1 1 1 1 1 1 þ þ þ þ þ þ þ ; 2 1 4 3 2n 2n 1 1 1 1 1 1 1 þ þ þ þ þ þ : 1 2 4 3 5 7 C09 12/08/2010 15:9:16 Page 269 9.1 ABSOLUTE CONVERGENCE 269 The first rearrangement is obtained from the harmonic series by interchanging the first and second terms, the third and fourth terms, and so forth. The second rearrangement is obtained from the harmonic series by taking one ‘‘odd term,’’ two ‘‘even terms,’’ three ‘‘odd terms,’’ and so forth. It is obvious that there are infinitely many other possible rearrangements of the harmonic series. P P 9.1.4 Definition A series yk in R is a rearrangement of a series xn if there is a bijection f of N onto N such that yk ¼ xf (k) for all k 2 N. While a grouping series does not affect the convergence of a series, making rearrangeP ments may do so. If fact, there is a remarkable observation, due to Riemann, that if sn is a conditionally convergent series in R, and if c 2 R is arbitrary, then there is a rearrangement P of xn that converges to c. To prove this assertion, we first note that a conditionally convergent series must contain infinitely many positive terms and infinitely many negative terms (see Exercise 1), and that both the series of positive terms and the series of negative terms diverge (see Exercise 2). To construct a series converging to c, we take positive terms until the partial sum is greater than c, then we take negative terms until the partial sum is less than c, then we take positive terms until the partial sum is greater than c, then we take negative terms, etc. In our manipulations with series, we generally want to be sure that rearrangements will not affect the convergence or the value of the series. That is why the following result is important. P 9.1.5 Rearrangement Theorem Let xn be an absolutely convergent series in R. P P Then any rearrangement yk of xn converges to the same value. P Proof. Suppose that xn converges to x 2 R. Thus, if e > 0, let N be such that if n, q > N and sn :¼ x1 þ þ xn, then jx sn j < e q X and jxk j < e : k¼Nþ1 Let M 2 N be such that all of the terms x1, . . . , xN are contained as summands in tM :¼ y1 þ þ yM. It follows that if m M, then tm sn is the sum of a finite number of terms xk with index k > N. Hence, for some q > N, we have jtm sn j q X jxk j < e: k¼Nþ1 Therefore, if m M, then we have jtm xj jtm sn j þ jsn xj < e þ e ¼ 2e: P Since e > 0 is arbitrary, we conclude that yk converges to x. Q.E.D. Exercises for Section 9.1 1. Show that if a convergent series contains only a finite number of negative terms, then it is absolutely convergent. 2. Show that if a series is conditionally convergent, then the series obtained from its positive terms is divergent, and the series obtained from its negative terms is divergent. C09 12/08/2010 15:9:16 270 Page 270 CHAPTER 9 INFINITE SERIES P 3. If an is conditionally convergent, give an argument to show that there exists a rearrangement whose partial sums diverge to 1. P 4. Where is the fact that the series xn is absolutely convergent used in the proof of 9.1.5? P P 5. If an is absolutely convergent, is it true that every rearrangement of an is also absolutely convergent? 1 X ln 1 1=n2 to show that this series 6. Find an explicit expression for the nth partial sum of converges to ln 2. Is this convergence absolute? n¼2 P P an bn is 7. (a) If an is absolutely convergent and (bn) is a bounded sequence, show that absolutely convergent. P (b) Give an example to show an is conditional and (bn) is a Pthat if the convergence of bounded sequence, then an bn may diverge. P P 8. Give an example of a convergent series an such that a2n is not convergent. (Compare this with Exercise 3.7.11.) P 9. If (an) is a decreasing sequence of strictly positive numbers and if an is convergent, show that lim(nan) ¼ 0. P 10. Give an example of a divergent series an with (an) decreasing and such that lim(nan) ¼ 0. P 11. If (an) is a sequence and if lim(n2an) exists in R, show that an is absolutely convergent. P n 1 12. Let a > 0. Show that the series ð1 þ a Þ is divergent if 0 < a 1 and is convergent if a > 1. pffiffiffi 1 pffiffiffiffiffiffiffiffiffiffiffi X nþ1 n pffiffiffi converge? 13. (a) Does the series n n¼1 pffiffiffi 1 pffiffiffiffiffiffiffiffiffiffiffi X nþ1 n converge? (b) Does the series n n¼1 P P 14. P If ðank Þ is a subsequence of (an), then the series ank is called a subseries of an . Show that an is absolutely convergent if and only if every subseries of it is convergent. 1 1 X X 15. Let a : N N ! R and write aij :¼ a(i, j). If Ai :¼ aij for each i 2 N and if A :¼ Ai , we j¼1 1 1 i¼1 XX say that A is an iterated sum of the aij and write A ¼ aij . We define the other iterated 1 X 1 X i¼1 j¼1 aij, in a similar way. sum, denoted by j¼1 i¼1 Suppose aij 0 for i; j 2 N. If (ck) is any enumeration of faij : i; j 2 Ng, show that the following statements are equivalent: 1 X 1 X (i) The iterated sum aij converges to B. 1 i¼1 j¼1 X ck converges to C. (ii) The series k¼1 In this case, we have B ¼ C. 16. The preceding exercise may fail if the terms are not positive. For example, let aij :¼ þ1 if i j ¼ 1, aij :¼ 1 if i j ¼ 1, and aij :¼ 0 elsewhere. Show that the iterated sums 1 X 1 1 X 1 X X aij and aij i¼1 j¼1 j¼1 i¼1 both exist but are not equal. Section 9.2 Tests for Absolute Convergence In Section 3.7 we gave some results concerning the convergence of infinite series; namely, the nth Term Test, the fact that a series of positive terms is convergent if and only if its C09 12/08/2010 15:9:17 Page 271 9.2 TESTS FOR ABSOLUTE CONVERGENCE 271 sequence of partial sums is bounded, the Cauchy Criterion, and the Comparison and Limit Comparison Tests. We will now give some additional results that may be familiar from calculus. These results are particularly useful in establishing absolute convergence. 9.2.1 Limit Comparison Test, II Suppose that X :¼ (xn) and Y :¼ (yn) are nonzero real sequences and suppose that the following limit exists in R : xn ð1Þ r :¼ lim : yn P P (a) If r 6¼ 0, then xn is absolutely convergent if and only if yn is absolutely convergent. P P xn is absolutely convergent. (b) If r ¼ 0 and if yn is absolutely convergent, then Proof. This result follows immediately from Theorem 3.7.8. Q.E.D. The Root and Ratio Tests The following test is due to Cauchy. 9.2.2 Root Test Let X :¼ (xn) be a sequence in R. (a) If there exist r 2 R with r < 1 and K 2 N such that ð2Þ jxn j1=n r P f or n K; then the series xn is absolutely convergent. (b) If there exists K 2 N such that jxn j1=n 1 ð3Þ then the series P f or n K; & xn is divergent. P Proof. (a) If (2) holds, then we have jxnj rn for n K. Since the geometric series rn P is convergent for 0 r < 1, the Comparison Test 3.7.7 implies that jxn j is convergent. (b) If (3) holds, then jxnj 1 for n K, so the terms do not approach 0 and the nth Term Test 3.7.3 applies. Q.E.D. In calculus courses, one often meets the following version of the Root Test. 9.2.3 Corollary Let X :¼ (xn) be a sequence in R and suppose that the limit ð4Þ exists in R. Then P r :¼ lim jxn j1=n xn is absolutely convergent when r < 1 and is divergent when r > 1. Proof. If the limit in (4) exists and r < 1, then there exist r1 with r < r1 < 1 and K 2 N such that jxnj1=n < r1 for n > K. In this case we can apply 9.2.2(a). If r > 1, then there exists K 2 N such that jxnj1=n > 1 for n K and the nth Term Test applies. Q.E.D. C09 12/08/2010 15:9:17 272 Page 272 CHAPTER 9 INFINITE SERIES Note No conclusion is possible in Corollary 9.2.3 when r ¼ 1, for either convergence or divergence is possible. See Example 9.2.7(b). Our next test is due to D’Alembert. 9.2.4 Ratio Test Let X :¼ (xn) be a sequence of nonzero real numbers. (a) If there exist r 2 R with 0 < r < 1 and K 2 N such that xnþ1 f or n K; ð5Þ x r n P then the series xn is absolutely convergent. (b) If there exists K 2 N such that xnþ1 f or ð6Þ x 1 n P then the series xn is divergent. n K; Proof. (a) If (5) holds, an Induction argument shows that jxKþmj jxKjrm for m 2 N. P Thus, for n K the in the P terms in jxn j are dominated by a fixed multiple of the termsP geometric series rm with 0 < r < 1. The Comparison Test 3.7.7 then implies that jxn j is convergent. (b) If (6) holds, an Induction argument shows that jxKþmj jxKj for m 2 N and the nth Term Test applies. Q.E.D. Once again we have a familiar result from calculus. 9.2.5 Corollary Let X :¼ (xn) be a nonzero sequence in R and suppose that the limit xnþ1 ð7Þ r :¼ lim xn P exists in R. Then xn is absolutely convergent when r < 1 and is divergent when r > 1. Proof. If r < 1 and if r < r1< 1, then there exists K 2 R such that jxnþ1P =xnj < r1 for n K. Thus Theorem 9.2.4(a) applies to give the absolute convergence of xn . If r > 1, then there exists K 2 N such that jxnþ1=xnj>1 for n K, whence it follows that jxkj does not converge to 0 and the nth Term Test applies. Q.E.D. Note No conclusion is possible in Corollary 9.2.5 when r ¼ 1, for either convergence or divergence is possible. See Example 9.2.7(c). The Integral Test The next test—a very powerful one—uses the notion of the improper Rintegral, which is b defined as follows: If f is in R[a, b] for every b > a and if the limit lim a f ðtÞdt exists in b!1 R1 R, then the improper integral a f ðtÞdt is defined to be this limit. C09 12/08/2010 15:9:17 Page 273 273 9.2 TESTS FOR ABSOLUTE CONVERGENCE 9.2.6 Integral Test Let f be a positive, decreasing function on {t : t 1}. Then the series 1 X f ðkÞ converges if and only if the improper integral k¼1 Z 1 Z b f ðtÞdt ¼ lim b!1 1 f ðtÞdt 1 exists. In the case of convergence, the partial sum sn ¼ n X f ðkÞ and the sum s ¼ k¼1 satisfy the estimate Z ð8Þ 1 nþ1 Z f ðtÞdt s sn 1 n 1 X f ðkÞ k¼1 f ðtÞdt: Proof. Since f is positive and decreasing on the interval [k 1, k], we have Z k ð9Þ f ðkÞ f ðtÞdt f ðk 1Þ: k1 By adding this inequality for k ¼ 2, 3, . . . , n, we obtain Z n sn f ð1Þ f ðtÞdt sn1 ; 1 which shows that either both or neither of the limits Z n lim sn and lim f ðtÞdt n!1 n!1 1 exist. If they exist, then on adding (9) for k ¼ n þ 1, . . . , m, we obtain Z m f ðtÞdt sm1 sn1 ; sm sn n from which it follows that Z mþ1 nþ1 Z f ðtÞdt sm sn m n f ðtÞdt: If we take the limit in this last inequality as m ! 1, we obtain (8). Q.E.D. We will now show how the results in Theorems 9.2.1–9.2.6 can be applied to the p-series, which were introduced in Example 3.7.6(d, e). P 2 9.2.7 Examples (a) Consider P the case p ¼ 2; that is, the series 1=n . We compare it with the convergent series 1=ðnðn þ 1ÞÞ of Example 3.7.2(c). Since 1 1 n þ 1 1 n2 nðn þ 1Þ ¼ n ¼ 1 þ n ! 1; P the Limit Comparison Test 9.2.1 implies that 1=n2 is convergent. (b) We demonstrate the failure of the Root Test for the p-series. Note that 1=n 1 ¼ 1 ¼ 1 p: np ðnp Þ1=n ðn1=n Þ Since (see Example 3.1.11(d)) we know that n1/n ! 1, we have r ¼ 1 in Corollary 9.2.3, and the theorem does not give any information. C09 12/08/2010 15:9:18 274 Page 274 CHAPTER 9 INFINITE SERIES (c) We apply the Ratio Test to the p-series. Since 1 1 np 1 ! 1; ¼ p p ¼ ðn þ 1Þ p n ð1 þ 1=nÞp ðn þ 1Þ the Ratio Test, in the form of Corollary 9.2.5, does not give any information. (d) Finally, we apply the Integral Test to the p-series. Let f (t) :¼ 1=tp for t 1 and recall that Z n 1 dt ¼ ln n ln 1; 1 t Z n 1 1 1 dt ¼ 1 for p 6¼ 1: p 1 p np1 1 t From these relations we see that the p-series converges if p > 1 and diverges if p 1, as we & have seen before in 3.7.6(d, e). Raabe’s Test If the limits limjxnj1=n and lim(jxnþ1=xnj) that are used in Corollaries 9.2.3 and 9.2.5 equal 1, we have seen that these tests do not give any information about the convergence or divergence of the series. In this case it is often useful to employ a more delicate test. Here is one that is frequently useful. 9.2.8 Raabe’s Test Let X :¼ (xn) be a sequence of nonzero real numbers. (a) If there exist numbers a > 1 and K 2 N such that xnþ1 a for n K; ð10Þ x 1n n P then xn is absolutely convergent. (b) If there exist real numbers a 1 and K 2 N such that xnþ1 a for n K; ð11Þ x 1n n P then xn is not absolutely convergent. & Proof. (a) If the inequality (10) holds, then we have (after replacing n by k and multiplying) kjxkþ1 j ðk 1Þjxk j ða 1Þjxk j for k K: On reorganizing the inequality, we have ð12Þ ðk 1Þjxk j kjxkþ1 j ða 1Þjxk j > 0 for k K; from which we deduce that the sequence (kjxkþ1j) is decreasing for k K. If we add (12) for k ¼ K, . . . , n and note that the left side telescopes, we get ðK 1ÞjxK j njxnþ1 j ða 1ÞðjxK j þ þ jxn jÞ: P This shows (why?) that the partial sums of jxn j are bounded and establishes the absolute convergence of the series. C09 12/08/2010 15:9:18 Page 275 9.2 TESTS FOR ABSOLUTE CONVERGENCE 275 (b) If the relation (11) holds for n K, then since a 1, we have njxnþ1 j ðn aÞjxn j ðn 1Þjxn j for nK: Therefore the sequence (njxnþ1j) is increasing for n K and there P exists a number c > 0 such that jx j > c=n for n K. But since the harmonic series 1=n diverges, the series nþl P jxn j also diverges. Q.E.D. In the application of Raabe’s Test, it is often convenient to use the following limiting form. 9.2.9 Corollary Let X :¼ (xn) be a nonzero sequence in R and let xnþ1 ; ð13Þ a :¼ lim n 1 xn P whenever this limit exists. Then xn is absolutely convergent when a > 1 and is not absolutely convergent when a < 1. Proof. Suppose the limit in (13) exists and that a > 1. If a1 is any number with a > a1 > 1, then there exists K 2 N such that a1 < n(1 jxnþ1=xnj) for n > K. Therefore jxnþ1=xnj < 1 a1=n for n K and Raabe’s Test 9.2.8(a) applies. The case where a < 1 is similar and is left to the reader. Q.E.D. Note There is no conclusion when a ¼ 1; either convergence or divergence is possible, as the reader can show. 9.2.10 Examples (a) We reconsider the p-series in the light of Raabe’s Test. Applying L’Hospital’s Rule when p 1, we obtain (why?) ð n þ 1Þ p np np a ¼ lim n 1 ¼ lim n p ð n þ 1Þ ð n þ 1Þ p ð1 þ 1=nÞp 1 1 ¼ lim lim ¼ p 1 ¼ p: 1=n ð1 þ 1=nÞp We conclude that if p > 1 then the p-series is convergent, and if 0 < p < 1 then the series is divergent (since the terms are positive). However, if p ¼ 1 (the harmonic series!), Corollary 9.2.9 yields no information. 1 P n : (b) We now consider 2 n¼1 n þ 1 An easy calculation shows that lim(xnþ1=xn) ¼ 1, so that Corollary 9.2.5 does not apply. Also, we have lim(n(1 xnþ1=xn)) ¼ 1, so that Corollary 9.2.9 does not apply either. However, it is an exercise to establish the inequality xnþ1=xn (n 1)=n, from which it follows from Raabe’s Test 9.2.8(b) that the series is divergent. (Of course, the Integral Test, & or the Limit Comparison Test with (yn) ¼ (1=n), can be applied here.) Although the limiting form 9.2.9 of Raabe’s Test is much easier to apply, Example 9.2.10(b) shows that the form 9.2.8 is stronger than 9.2.9. Remark Leonard Euler’s calculational prowess was remarkable, including his work in the area of infinite series. His methods are not covered in this book, but we will state one of his famous results. We know the series 112 þ 212 þ 312 þ þn12 þ converges C09 12/08/2010 15:9:19 276 Page 276 CHAPTER 9 INFINITE SERIES (Example 9.2.7(a)), but the problem of determining its exact value is quite difficult. The problem was known as the Basel problem, named after Basel University in Switzerland, 1 X 1 p2 . He also ¼ and Euler solved it in 1735 when he obtained the surprising result that n2 6 n¼1 described a process to derive the values of series with higher even powers of 1=n. For 1 1 X X 1 p4 1 p6 example, he showed that and . However, the values of the ¼ ¼ n4 90 n6 945 n¼1 n¼1 corresponding series of odd powers of 1=n were not discovered and stay elusive to the present day. Another famous problem involves a number that is defined as follows. The harmonic series is known to diverge, but Euler observed that the sequence 1 1 1 cn :¼ 1 þ þ þ þ ln n 2 3 n is convergent. (See Exercise 15.) The limit of the sequence, g :¼ lim (cn), is called Euler’s constant and is approximately equal to 0.5772156649. . . . The question of whether g is a rational or irrational number is a famous unsolved problem. Computers have calculated over two trillion decimal places, and though the prevailing opinion is that g is irrational, no proof has yet been found. Exercises for Section 9.2 1. Establish the convergence or the divergence of the series whose nth term is: (a) 1 , ðn þ 1Þðn þ 2Þ (c) 21=n , (b) n , ðn þ 1Þðn þ 2Þ (d) n=2n. 2. Establish the convergence or divergence of the series whose nth term is: (b) ðn2 ðn þ 1ÞÞ1=2 ; (a) ðnðn þ 1ÞÞ1=2 ; n (c) n!=n ; (d) ð1Þn n=ðn þ 1Þ: 3. Discuss the convergence or the divergence of the series with nth term (for sufficiently large n) given by (b) ðln nÞn ; (a) ðln nÞp ; (c) ðln nÞln n ; (d) ðln nÞln ln n ; 1 (e) ðn ln nÞ ; (f) ðnðln nÞðln ln nÞ2 Þ1 4. Discuss the convergence or the divergence of the series with nth term: (a) 2n en ; (b) nn en ; pffiffi ln n (c) e ; (d) ðln nÞe n ; 2 n (e) n!e ; (f) n!en : 5. Show that the series 1=12 þ 1=23 þ 1=32 þ 1=43 þ is convergent, but that both the Ratio and the Root Tests fail to apply. P 6. If a and b are positive numbers, then ðan þ bÞp converges if p > 1 and diverges if p 1. 7. Discuss the series whose nth term is n! ; 3 5 7 ð2n þ 1Þ 2 4 ð2nÞ ; (c) 3 5 ð2n þ 1Þ (a) (b) (d) ðn!Þ2 ; ð2nÞ! 2 4 ð2nÞ : 5 7 ð2n þ 3Þ C09 12/08/2010 15:9:20 Page 277 9.3 TESTS FOR NONABSOLUTE CONVERGENCE 277 8. Let 0 < a < 1 and consider the series a2 þ a þ a4 þ a3 þ þ a2n þ a2n1 þ : Show that the Root Test applies, but that the Ratio Test does not apply. 9. If r 2 ð0; 1Þ satisfies (2) in the Root Test 9.2.2, show that the partial sums sn of approximate its limit s according to the estimate js sn j rnþ1 =ð1 rÞ for n K. P xn 10. If r 2 (0, 1) satisfies (5) in the Ratio Test 9.2.4, show that js sn j rjxn j=ð1 rÞ for n K. 11. If a > 1 satisfies (10) in Raabe’s Test 9.2.8, show that js sn j njxn j=ða 1Þ for n K. 12. For each of the series in Exercise 1 that converge, estimate the remainder if only four terms are taken; if only ten terms are taken. If we wish to determine the sum of the series within 1=1000, how many terms should be taken? 13. Answer the questions posed in Exercise 12 for the series given in Exercise 2. 14. Show that the series 1 þ 12 13 þ 14 þ 15 16 þ þ is divergent. 15. For n 2 N, let cn be defined by cn :¼ 11 þ 12 þ þ 1=n ln n. Show that (cn) is a decreasing sequence of positive numbers. Show that if we put 1 1 1 1 þ ; 1 2 3 2n then the sequence (bn) converges to ln 2. [Hint: bn ¼ c2n cn þ ln 2.] bn :¼ 16. Let {n1, n2, . . .} denote the collection of natural numbers that do not use the digit 6 in their P decimal expansion. Show that 1=nk converges P to a number less than 80. If {m1, m2, . . . } is the collection of numbers that end in 6, then 1=mk diverges. If {p1, p2, . . . } is the collection P of numbers that do not end in 6, then 1=pk diverges. 17. If p > 0, q > 0, show that the series X ðp þ 1Þðp þ 2Þ ðp þ nÞ ðq þ 1Þðq þ 2Þ ðq þ nÞ converges for q > p þ 1 and diverges for q p þ 1. 18. Suppose that none of the numbers a, b, c is a negative integer or zero. Prove that the hypergeometric series ab aða þ 1Þbðb þ 1Þ aða þ 1Þða þ 2Þbðb þ 1Þðb þ 2Þ þ þ þ 1!c 2!cðc þ 1Þ 3!cðc þ 1Þðc þ 2Þ is absolutely convergent for c > a þ b and divergent for c < a þ b. P P 19. Let an > 0 and suppose P that an converges. Construct a convergent series bn with bn > 0 such that P lim(a Let (An) be the partial sums of pffiffiffi lessprapidly ffiffiffiffiffiffiffiffiffiffiffiffiffiffithan an . [Hint: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi P n=bn) ¼ 0; hence bn converges an and A its limit. Define b1 :¼ A A A1 and bn :¼ A An1 A An for n 1.] P 20. Let (an) be a decreasing sequence an diverges. P of real numbers converging to 0 and suppose that P Construct a divergent series b with b > 0 such that lim(b =a ) ¼ 0; hence b diverges less n n npffiffiffiffiffi n n P P rapidly than an . [Hint: Let bn :¼ an = An where An is the nth partial sum of an .] Section 9.3 Tests for Nonabsolute Convergence The convergence tests that were discussed in the preceding section were primarily directed to establishing the absolute convergence of a series. Since there are many series, such as ð1Þ 1 X ð1Þnþ1 n¼1 n ; 1 X ð1Þnþ1 pffiffiffi ; n n¼1 C09 12/08/2010 15:9:20 278 Page 278 CHAPTER 9 INFINITE SERIES that are convergent but not absolutely convergent, it is desirable to have some tests for this phenomenon. In this short section we shall present first the test for alternating series and then tests for more general series due to Dirichlet and Abel. Alternating Series The most familiar test for nonabsolutely convergent series is the one due to Leibniz that is applicable to series that are ‘‘alternating’’ in the following sense. 9.3.1 Definition A sequence X :¼ (xn) of nonzero real numbers is said to be alternating if the terms ð1Þnþ1 xn ; n 2 N, are all positive (or Pall negative) real numbers. If the sequence X ¼ (xn) is alternating, we say that the series xn it generates is an alternating series. In the case of an alternating series, it is useful to set xn ¼ (1)nþ1zn [or xn ¼ (1)nzn], where zn > 0 for all n 2 N. 9.3.2 Alternating Series Test Let Z :¼ (zn) be a decreasing of strictly posiP sequence tive numbers with limðzn Þ ¼ 0. Then the alternating series ð1Þnþ1 zn is convergent. Proof. Since we have s2n ¼ ðz1 z2 Þ þ ðz3 z4 Þ þ þ ðz2n1 z2n Þ; and since zk zkþ1 0, it follows that the subsequence (s2n) of partial sums is increasing. Since s2n ¼ z1 ðz2 z3 Þ ðz2n2 z2n1 Þ z2n ; it also follows that s2n z1 for all n 2 N. It follows from the Monotone Convergence Theorem 3.3.2 that the subsequence (s2n ) converges to some number s 2 R. We now show that the entire sequence (sn) converges to s. Indeed, if e > 0, let K be such that if n K then js2n sj 12 e and jz2nþ1 j 12 e. It follows that if n K then js2nþ1 sj ¼ js2n þ z2nþ1 sj js2n sj þ jz2nþ1 j 12 e þ 12 e ¼ e: Therefore every partial sum of an odd number of terms is also within e of is large Ps if n nþ1 ð1Þ zn is enough. Since e > 0 is arbitrary, the convergence of (sn) and hence of established. Q.E.D. Note It is an exercise to show that if s is the sum of the alternating series and if sn is its nth partial sum, then ð2Þ js sn j znþ1 : It is clear that this Alternating Series Test establishes the convergence of the two series already mentioned, in (1). The Dirichlet and Abel Tests We will now present two other tests of wide applicability. They are based on the following lemma, which is sometimes called the partial summation formula, since it corresponds to the familiar formula for integration by parts. C09 12/08/2010 15:9:21 Page 279 9.3 TESTS FOR NONABSOLUTE CONVERGENCE 279 9.3.3 Abel’s P Lemma Let X :¼ (xn) and Y :¼ (yn) be sequences in R and let the partial sums of yn be denoted by (sn) with s0 :¼ 0. If m > n, then m m1 X X xk yk ¼ ðxm sm xnþ1 sn Þ þ ðxk xkþ1 Þsk : ð3Þ k¼nþ1 k¼nþ1 Proof. Since yk ¼ sk sk1 for k ¼ 1, 2, . . . , the left side of (3) is seen to be equal to m X xk ðsk sk1 Þ. If we collect the terms multiplying sn, snþ1, . . . , sm, we obtain the k¼nþ1 right side of (3). Q.E.D. P We now apply Abel’s Lemma to obtain tests for convergence of series of the form xn yn . 9.3.4 Dirichlet’s Test P If X :¼ (xn) is a decreasing sequence P with lim xn ¼ 0, and if the partial sums (sn) of yn are bounded, then the series xn yn is convergent. Proof. Let jsn j B for all n 2 N. If m > n, it follows from Abel’s Lemma 9.3.3 and the fact that xk xkþ1 0 that X m m1 X xk yk ðxm þ xnþ1 ÞB þ ðxk xkþ1 ÞB k¼nþ1 k¼nþ1 ¼ ½ðxm þ xnþ1 Þ þ ðxnþ1 xm ÞB ¼ 2xnþ1 B: P Since lim(xk) ¼ 0, the convergence of xk yk follows from the Cauchy Convergence Criterion 3.7.4. Q.E.D. 9.3.5 Abel’s Test If X :¼ (xnP ) is a convergent monotone sequence and the series is convergent, then the series xn yn is also convergent. P yn Proof. If (xn) is decreasing with limit x, let un :¼ xn x, n 2 N, so that (un) decreases to 0. Then xn ¼ x þ un, whence xnyn P ¼ xyn þ unyn. It follows from the Dirichlet Test 9.3.4 that P and, since xy un yn is convergent P Pn converges (because of the assumed convergence of the series yn ), we conclude that xn yn is convergent. If (xn) is increasing with limit x, let vn :¼ x xn, n 2 N, so that (vn) decreases to 0. Here xn ¼ x vn, whence xnyn ¼ xyn vnyn, and the argument proceeds as before. Q.E.D. 9.3.6 Examples (a) Since we have 2ðsin 12 xÞ ðcos x þ þ cos nxÞ ¼ sinðn þ 12Þx sin 12 x; it follows that if x 6¼ 2kp ðk 2 NÞ, then jcos x þ þ cos nxj ¼ jsinðn þ 12Þx sin 12 xj 1 : 1 j2 sin 2 xj jsin 12 xj Hence Dirichlet’s Test implies that if (an) is decreasing with lim(an) ¼ 0, then the series 1 X an cos nx converges provided x 6¼ 2kp. n¼1 C09 12/08/2010 15:9:21 280 Page 280 CHAPTER 9 INFINITE SERIES (b) Since we have 2 sin 12 x ðsin x þ þ sin nxÞ ¼ cos 12 x cos n þ 12 x; it follows that if x 6¼ 2kp ðk 2 NÞ, then jsin x þ þ sin nxj 1 : sin 12 x 1 X As before, if (an) is decreasing and if lim(an) ¼ 0, then the series an sin nx converges for n¼1 & x 6¼ 2kp (and it also converges for these values). Exercises for Section 9.3 1. Test the following series for convergence and for absolute convergence: 1 ð1Þnþ1 1 ð1Þnþ1 P P (a) ; (b) ; 2 n¼1 n þ 1 n¼1 n þ 1 1 ð1Þnþ1 n 1 P P ln n ð1Þnþ1 (c) ; (d) : n þ 2 n n¼1 n¼1 1 X ð1Þnþ1 zn , and if s denotes the sum of this 2. If sn is the nth partial sum of the alternating series n¼1 series, show that js sn j znþ1 . 3. Give an example to show that the Alternating Series Test 9.3.2 may fail if (zn) is not a decreasing sequence. 4. Show that the Alternating Series Test is a consequence of Dirichlet’s Test 9.3.4. 5. Consider the series 1 1 1 1 1 1 1 þ þ þ þ ; 2 3 4 5 6 7 where the signs come in pairs. Does it converge? P an =np is convergent, show that the series 6. Let P an 2q R for n 2 N and let p < q. If the series an =n is also convergent. P 7. If p and q are positive numbers, show that ð1Þn ðln nÞp =nq is a convergent series. 8. Discuss the series whose nth term is: nn (a) ð1Þn ; ðn þ 1Þnþ1 nn ðn þ 1Þnþ1 ; ðn þ 1Þn : nnþ1 1 X P an ent converges for t > 0. 9. If the partial sums of an are bounded, show that the series 1 1 X X n¼1 10. If the partial sums sn of an are bounded, show that the series an =n converges to 1 n¼1 n¼1 X sn =nðn þ 1Þ. (c) ð1Þn ðn þ 1Þn ; nn (b) (d) n¼1 11. Can Dirichlet’s Test be applied to establish the convergence of 1 1 1 1 1 1 þ þ þ 2 3 4 5 6 where the number of signs increases by one in each ‘‘block’’? If not, use another method to establish the convergence of this series. C09 12/08/2010 15:9:22 Page 281 9.4 SERIES OF FUNCTIONS 281 12. Show that the hypothesis that the sequence X :¼ (xn) is decreasing in Dirichlet’s Test 9.3.4 can 1 X be replaced by the hypothesis that jxn xnþ1 j is convergent. n¼1 13. If (an) is a bounded decreasing sequence and (bn) is a bounded increasing sequence and if xn :¼ 1 X an þ bn for n 2 N, show that jxn xnþ1 j is convergent. 1 n¼1 X ak satisfy jsn j Mnr for some r < 1, then the 14. Show that if the partial sums sn of the series 1 X k¼1 series an =n converges. n¼1 P P 15. Suppose that an is a convergent series of real numbers. Either prove that bn converges or give a counter-example, when we define bn by pffiffiffiffiffi (a) an =n, (b) an =n ðan 0Þ, pffiffiffiffiffiffiffiffiffiffi (c) an sin n, (d) an =n ðan 0Þ, (e) n1=n an , (f) an =ð1 þ jan jÞ. Section 9.4 Series of Functions Because of their frequent appearance and importance, we now present a discussion of infinite series of functions. Since the convergence of an infinite series is handled by examining the sequence of partial sums, questions concerning series of functions are answered by examining corresponding questions for sequences of functions. For this reason, a portion of the present section is merely a translation of facts already established for sequences of functions into series terminology. However, in the second part of the section, where we discuss power series, some new features arise because of the special character of the functions involved. 9.4.1 Definition If ( fn) is a sequence of functions defined on Pa subset D of R with values in R, the sequence of partial sums (sn) of the infinite series f n is defined for x in D by s1 ðxÞ :¼ f 1 ðxÞ; s2 ðxÞ :¼ s1 ðxÞ þ f 2 ðxÞ . . . . . . . .. . . . . . . . . . . . . . . . . . snþ1 ðxÞ :¼ sn ðxÞ þ f nþ1 ðxÞ . . . . . . . .. . . . . . . . . . . . . . . . . . In case the sequenceP (sn) of functions converges on D to a function f, we say that the infinite series of functions f n converges to f on D. We will often write 1 X X fn or fn n¼1 to denote either the series or the limit function, when it exists. P P If the series j f n ðxÞj converges for each x in D, we say that f n is absolutely convergent on D. If the sequence (s ) of partial sums is uniformly convergent on D to f, we n P say that f n is uniformly convergent on D, or that it converges to f uniformly on D. One of the main reasons for the interest in uniformly convergent series of functions is the validity of the following results, which give conditions justifying the change of order of the summation and other limiting operations. 9.4.2 Theorem If fn is continuous on D R to R for each n 2 N and if to f uniformly on D, then f is continuous on D. P f n converges C09 12/08/2010 15:9:22 282 Page 282 CHAPTER 9 INFINITE SERIES This is a direct translation of Theorem 8.2.2 for series. The next result is a translation of Theorem 8.2.4. 9.4.3 Theorem Suppose that the real-valued P functions fn , n 2 N, are Riemann integrable on the interval J :¼ [a, b]. If the series f n converges to f uniformly on J, then f is Riemann integrable and Z b 1 Z b X ð1Þ f ¼ f n: a n¼1 a Next we turn to the corresponding theorem pertaining to differentiation. Here we assume the uniform convergence of the series obtained after term-by-term differentiation of the given series. This result is an immediate consequence of Theorem 8.2.3. 9.4.4 Theorem For each n 2 N , let fn beP a real-valued junction on J :¼ [a, b] that has a derivative f 0n on J. Suppose that the series f n converges for at least one point of J and P 0 that the series of derivatives f n converges uniformly on P J. Then there exists a real-valued function f on J such that f n converges uniformly on P 0 J to f. In addition, f has a derivative on J and f 0 ¼ f n. Tests for Uniform Convergence Since we have stated some consequences of uniform convergence of series, we shall now present a few tests that can be used to establish uniform convergence. 9.4.5 P Cauchy Criterion Let ( fn) be a sequence of functions on D R to R. The series f n is uniformly convergent on D if and only if for every e > 0 there exists an M(e) such that if m > n M(e), then j f nþ1 ðxÞ þ þ f m ðxÞj < e for all x 2 D: of positive real numbers such that 9.4.6 Weierstrass M-Test Let (Mn) be a sequence P P j f n ðxÞj M n for x 2 D, n 2 N. If the series M n is convergent, then f n is uniformly convergent on D. Proof. If m > n, we have the relation j f nþ1 ðxÞ þ þ f m ðxÞj M nþ1 þ þ M m P Now apply 3.7.4, 9.4.5, and the convergence of Mn. for x 2 D: Q.E.D. In Appendix E we will use the Weierstrass M-Test to construct two interesting examples. Power Series We shall now turn to a discussion of power series. This is an important class of series of functions and enjoys properties that are not valid for general series of functions. 9.4.7 Definition A series of real functions x ¼ c if the function fn has the form P f n is said to be a power series around f n ðxÞ ¼ an ðx cÞn ; where an and c belong to R and where n ¼ 0, 1, 2, . . . . C09 12/08/2010 15:9:22 Page 283 9.4 SERIES OF FUNCTIONS 283 For the sake of simplicity of our notation, we shall treat only the case where c ¼ 0. This is no loss of generality, however, since the translation x0 ¼ x c reduces a power series around c to a power series around 0. Thus, whenever we refer to a power series, we shall mean a series of the form ð2Þ 1 X an xn ¼ a0 þ a1 x þ þ an xn þ : n¼0 Even though the functions appearing in (2) are defined over all of R, it is not to be expected that the series (2) will converge for all x in R. For example, by using the Ratio Test 9.2.4, we can show that the series 1 1 X X n!xn ; xn ; 1 X xn =n!; n¼0 n¼0 n¼0 converge for x in the sets f0g; fx 2 R : jxj < 1g; R; respectively. Thus, the set on which a power series converges may be small, medium, or large. However, an arbitrary subset of R cannot be the precise set on which a power series converges, as we shall show. If (bn) is a bounded sequence of nonnegative real numbers, then we define the limit superior of (bn) to be the infimum of those numbers v such that bn v for all sufficiently large n 2 N. This infimum is uniquely determined and is denoted by lim sup(bn). The only facts we need to know are (i) that if v > lim sup(bn), then bn v for all sufficiently large n 2 N, and (ii) that if w < lim sup(bn), then w bn for infinitely many n 2 N. (See 3.4.10 and 3.4.11.) P 9.4.8 Definition Let an xn be a power series. If the sequence ðjan j1=n Þ is bounded, we set r :¼ lim supðjan j1=n Þ; ifP this sequence is not bounded we set r ¼ þ1. We define the radius of convergence of an xn to be given by 8 if r ¼ þ1; <0 R :¼ 1=r if 0 < r < þ1; : þ1 if r ¼ 0: The interval of convergence is the open interval (R, R). We shall now justify the term ‘‘radius of convergence.’’ 9.4.9 Cauchy-Hadamard Theorem If R is the radius of convergence of the power P series an xn , then the series is absolutely convergent if jxj < R and is divergent if jxj > R. Proof. We shall treat only the case where 0 < R < þ1, leaving the cases R ¼ 0 and R ¼ þ1 as exercises. If 0 < jxj < R, then there exists a positive number c < 1 such that jxj < cR. Therefore r < c=jxj and so it follows that if n is sufficiently large, then jan j1=n c=jxj. This is equivalent to the statement that ð3Þ jan xn j cn for all sufficiently large n. Since c < 1, the absolute convergence of the Comparison Test 3.7.7. P an xn follows from C09 12/08/2010 15:9:22 284 Page 284 CHAPTER 9 INFINITE SERIES If jxj > R ¼ 1=r, then there are infinitely many n 2 N for which jan j1=n > 1=jxj. Therefore, jan xn j > 1 for infinitely many n, so that the sequence ðan xn Þ does not converge to zero. Q.E.D. Remark It will be noted that the Cauchy-Hadamard Theorem makes no statement as to whether the power series converges when jxj ¼ R. Indeed, anything can happen, as the examples X X1 X1 xn ; xn ; xn ; n n2 show. Since lim(n1/n) ¼ 1, each of these power series has radius of convergence equal to 1. The first power series converges at neither of the points x ¼ 1 and x ¼ þ1; the second series converges at x ¼ 1 but diverges at x ¼ þ1; and the third power series converges at both x ¼ 1 and x ¼ þ1. (Find a power series with R ¼ 1 that converges at x ¼ þ1 but diverges x ¼ 1.) P It is an exercise to show that the radius of convergence of the series an xn is also given by an ; ð4Þ lim anþ1 provided this limit exists. Frequently, it is more convenient to use (4) than Definition 9.4.8. The argument used in the proof of the Cauchy-Hadamard Theorem yields the uniform convergence of the power series on any fixed closed and bounded interval in the interval of convergence (R, R). P 9.4.10 Theorem Let R be the radius of convergence of an xn and let K be a closed and bounded interval contained in the interval of convergence (R, R). Then the power series converges uniformly on K. Proof. The hypothesis on K ðR; RÞ implies that there exists a positive constant c < 1 such that jxj < cR for all x 2 K. (Why?) By the argument in 9.4.9, we infer that for sufficiently largePn, the estimate (3) holds for all x 2 K. Since c < 1, the uniform convergence of an xn on K is a direct consequence of the Weierstrass M-test with M n :¼ cn . Q.E.D. 9.4.11 Theorem The limit of a power series is continuous on the interval of convergence. A power series can be integrated term-by-term over any closed and bounded interval contained in the interval of convergence. P Proof. If jx0 j < R, then the preceding result asserts that an xn converges uniformly on any closed and bounded neighborhood of x0 contained in (R, R). The continuity at x0 then follows from Theorem 9.4.2, and the term-by-term integration is justified by Theorem 9.4.3. Q.E.D. We now show that a power series can be differentiated term-by-term. Unlike the situation for general series, we do not need to assume that the differentiated series is uniformly convergent. Hence this result is stronger than Theorem 9.4.4. C09 12/08/2010 15:9:22 Page 285 9.4 SERIES OF FUNCTIONS 285 9.4.12 Differentiation Theorem A power series can be differentiated term-by-term within the interval of convergence. In fact, if 1 1 X X an xn ; then f 0 ðxÞ ¼ nan xn1 for jxj < R: f ðxÞ ¼ n¼0 n¼1 Both series have the same radius of convergence. Proof. Since lim(n1=n) ¼ 1, the sequence ðjnan j1=n Þ is bounded if and only if the sequence ðjan j1=n Þ is bounded. Moreover, it is easily seen that lim sup jnan j1=n ¼ lim sup jan j1=n : Therefore, the radius of convergence of the two series is the same, so the formally differentiated series is uniformly convergent on each closed and bounded interval contained in the interval of convergence. We can then apply Theorem 9.4.4 to conclude that the formally differentiated series converges to the derivative of the given series. Q.E.D. Remark It is to be observed that the theorem makes no assertion about the endpoints of the interval of convergence. If a series is convergent at an endpoint, then the differentiated 1 X series may or may not be convergent at this point. For example, the series xn =n2 n¼1 converges at both endpoints x ¼ 1 and x ¼ þ1. However, the differentiated series given 1 X by xn1 =n converges at x ¼ 1 but diverges at x ¼ þ1. n¼1 1 X By repeated application of the preceding result, we conclude that if k 2 N then an xn can be differentiated term-by-term k times to obtain n¼0 1 X n! an xnk : ð5Þ ð n k Þ! n¼k Moreover, this series converges absolutely to f ðkÞ ðxÞ for jxj < R and uniformly over any closed and bounded interval in the interval of convergence. If we substitute x ¼ 0 in (5), we obtain the important formula f ðkÞ ð0Þ ¼ k!ak : P P bn xn converge on some interval 9.4.13 Uniqueness Theorem If an xn and (r, r), r > 0, to the same function f, then an ¼ bn for all n2N: Proof. Our preceding remarks show that n!an ¼ f ðnÞ ð0Þ ¼ n!bn for all n 2 N. Q.E.D. Taylor Series If a function f has derivatives of all orders at a point c in R, then we can calculate the Taylor coefficients by a0 :¼ f ðcÞ; an :¼ f ðnÞ ðcÞ=n! for n 2 N and in this way obtain a power series with these coefficients. However, it is not necessarily true that the resulting power series converges to the function f in an interval about c. (See Exercise 12 for an example.) The issue of convergence is resolved by the remainder term Rn in Taylor’s Theorem 6.4.1. We will write ð6Þ f ðxÞ ¼ 1 ðnÞ X f ðcÞ n¼0 n! ðx cÞn C09 12/08/2010 15:9:23 286 Page 286 CHAPTER 9 INFINITE SERIES for jx cj < R if and only if the sequence (Rn (x)) of remainders converges to 0 for each x in some interval fx : jx cj < Rg. In this case we say that the power series (6) is the Taylor expansion of f at c. We observe that the Taylor polynomials for f discussed in Section 6.4 are just the partial sums of the Taylor expansion (6) of f. (Recall that 0! :¼ 1.) 9.4.14 Examples (a) If f (x) :¼ sin x, x 2 R, we have f (2n)(x) ¼ (1)n sin x and f (2nþ1)(x) ¼ (1)n cos x for n 2 N, x 2 R. Evaluating at c ¼ 0, we get the Taylor coefficients a2n ¼ 0 and a2nþ1 ¼ (l)n=(2n þ 1)! for n 2 N. Since jsin xj 1 and jcos xj 1 for all x, then jRn ðxÞj jxjn =n! for n 2 N and x 2 R. Since lim(Rn(x)) ¼ 0 for each x 2 R, we obtain the Taylor expansion sin x ¼ 1 X ð1Þn 2nþ1 x ð2n þ 1Þ! n¼0 for all x 2 R: An application of Theorem 9.4.12 gives us the Taylor expansion cos x ¼ 1 X ð1Þn n¼0 ð2nÞ! x2n for all x 2 R: (b) If gðxÞ :¼ ex , x 2 R, then gðnÞ ðxÞ ¼ ex for all n 2 N, and hence the Taylor coefficients are given by an ¼ 1=n! for n 2 N. For a given x 2 R, we have jRn ðxÞj ejxj jxjn =n! and therefore (Rn(x)) tends to 0 as n ! 1. Therefore, we obtain the Taylor expansion ex ¼ ð7Þ 1 X 1 n x n! n¼0 for all x2R: We can obtain the Taylor expansion at an arbitrary c 2 R by the device of replacing x by x c in (7) and noting that ex ¼ ec exc ¼ ec 1 1 c X X 1 e ð x cÞ n ¼ ð x cÞ n n! n! n¼0 n¼0 for x 2 R: & Exercises for Section 9.4 P 1. Discuss the convergence and the uniform convergence of the series f n , where f n ðxÞ is given by: 1 (a) ðx2 þ n2 Þ ; (b) ðnxÞ2 ðx 6¼ 0Þ ; (d) ðxn þ 1Þ1 ðx 6¼ 0Þ; (c) sinðx=n2 Þ; n n (e) x =ðx þ 1Þ ðx 0Þ; (f) ð1Þn ðn þ xÞ1 ðx 0Þ: P P an sin nx is absolutely and 2. If an is an absolutely convergent series, then the series uniformly convergent. P 3. Let (cn) be a decreasing sequence of positive numbers. If cn sin nx is uniformly convergent, then limðncn Þ ¼ 0. 4. Discuss the cases R ¼ 0, R ¼ þ1 in the Cauchy-Hadamard Theorem 9.4.9. P 5. Show that the radius of convergence R of the power series an xn is given by limðjan =anþ1 jÞ whenever this limit exists. Give an example of a power series where this limit does not exist. P 6. Determine the radius of convergence of the series an xn , where an is given by: n (a) 1=n , (b) na =n!, (c) nn =n!, (d) ðlnpffiffinÞ1 ; n 2 , 2 (e) ðn!Þ =ð2nÞ!, (f) n n . C09 12/08/2010 15:9:24 Page 287 9.4 SERIES OF FUNCTIONS 287 7. If an :¼ 1 when P n is the square of a natural number and an :¼ 0 otherwise, find the radius of convergence of an xn . If bn :¼ 1 when P nn ¼ m! for m 2 N and bn :¼ 0 otherwise, find the radius of convergence of the series bn x . 1=n 8. Prove in detail that lim sup jnan j ¼ lim sup jan j1=n . P an xn . 9. If 0 < p jan j q for all n 2 N, find the radius of convergence of P n 10. Let f ðxÞ ¼ an x for jxj < R. If f ðxÞ ¼ f ðxÞ for all jxj < R, show that an ¼ 0 for all odd n. 11. Prove that if f is defined for jxj < r and if there exists a constant B such that j f ðnÞ ðxÞj B for all jxj < r and n 2 N, then the Taylor series expansion 1 X f ðnÞ ð0Þ n x n! n¼0 converges to f (x) for jxj < r. 12. Prove by Induction that the function given by f ðxÞ :¼ e1=x for x 6¼ 0, f ð0Þ :¼ 0, has derivatives of all orders at every point and that all of these derivatives vanish at x ¼ 0. Hence this function is not given by its Taylor expansion about x ¼ 0. 2 13. Give an example of a function that is equal to its Taylor series expansion about x ¼ 0 for x 0, but is not equal to this expansion for x < 0. 14. Use the Lagrange form of the remainder to justify the general Binomial Expansion 1 X m n m for 0 x < 1 : ð1 þ xÞ ¼ x n n¼0 15. (Geometric series) Show directly that if jxj < 1, then 1=ð1 xÞ ¼ 1 X 16. Show by integrating the series for 1=(1 þ x) that if jxj < 1, then lnð1 þ xÞ ¼ 1 X ð1Þnþ1 n¼1 n n xn : 17. Show that if jxj < 1, then Arctan x ¼ 1 X ð1Þ 2nþ1 x . 2n þ1 n¼0 18. Show that if jxj < 1, then Arcsin x ¼ 1 X 1 3 ð2n 1Þ x2nþ1 . 2 4 2n 2n þ 1 n¼0 Z x et dt for x 2 R. Z a 1 k2 ðsin xÞ2 20. If a 2 R and jkj < 1, the integral F ða; kÞ :¼ 0 integral of the first kind. Show that 1 p pX 1 3 ð2n 1Þ 2 2n k for F ;k ¼ 2 2 n¼0 2 4 2n 19. Find a series expansion for 0 xn . n¼0 2 1=2 dx is called an elliptic jkj < 1 : C10 12/09/2010 15:0:8 Page 288 CHAPTER 10 THE GENERALIZED RIEMANN INTEGRAL In Chapter 7 we gave a rather complete discussion of the Riemann integral of a function on a closed bounded interval, defining the integral as the limit of Riemann sums of the function. This is the integral (and the approach) that the reader met in calculus courses; it is also the integral that is most frequently used in applications to engineering and other areas. We have seen that continuous and monotone functions on [a, b] are Riemann integrable, so most of the functions arising in calculus are included in its scope. However, by the end of the nineteenth century, some inadequacies in the Riemann theory of integration had become apparent. These failings came primarily from the fact that the collection of Riemann integrable functions became inconveniently small as mathematics developed. For example, the set of functions for which the Newton–Leibniz formula, Z b F 0 ¼ FðbÞ FðaÞ; a holds does not include all differentiable functions. Also, limits of sequences of Riemann integrable functions are not necessarily Riemann integrable. These inadequacies led others to invent other integration theories, the best known of which was due to Henri Lebesgue (1875–1941) and was developed at the very beginning of the twentieth century. (For an account of the history of the development of the Lebesgue integral, the reader should consult the book by Hawkins given in the References.) Indeed, the Lebesgue theory of integration has become pre-eminent in contemporary mathematical research, since it enables one to integrate a much larger collection of functions, and to take limits of integrals more freely. However, the Lebesgue integral also has several inadequacies and difficulties: (1) There exist functions F that are differentiable on [a, b] but such that F 0 is not Lebesgue integrable. (2) Some ‘‘improper integrals,’’ such as the important Dirichlet integral, Z 1 sin x dx; x 0 do not exist as Lebesgue integrals. (3) Most treatments of the Lebesgue integral have considerable prerequisites and are not easily within the reach of an undergraduate student of mathematics. As important as the Lebesgue integral is, there are even more inclusive theories of integration. One of these was developed independently in the late 1950s by the Czech mathematician Jaroslav Kurzweil (b. 1926) and the English mathematician Ralph Henstock (b. 1923). Surprisingly, their approach is only slightly different from that used by Riemann, yet it yields an integral (which we will call the generalized Riemann integral ) that includes both the Riemann and the Lebesgue integrals as special cases. Since the approach is so similar to that of Riemann, it is technically much simpler than the usual 288 C10 12/09/2010 15:0:8 Page 289 10.1 DEFINITION AND MAIN PROPERTIES 289 Ralph Henstock and Jaroslav Kurzweil Ralph Henstock (1923–2007), pictured on the left, was born in Nottinghamshire, England, the son of a mineworker. At an early age he showed that he was a gifted scholar in mathematics and science. He entered St. John’s College, Cambridge, in 1941, studying with J. D. Bernal, G. H. Hardy, and J. C. Burkhill and was classified Wrangler in Part II of the Tripos Exams in 1943. He earned his B.A. at Cambridge in 1944 and his Ph.D. at the University of London in 1948. His research is in the theory of summability, linear analysis, and integration theory. Most of his teaching was in Northern Ireland. Jaroslav Kurzweil (pictured on the right) was born on May 7, 1926, in Prague. A student of V. Jarnik, he has done a considerable amount of research in the theory of differential equations and the theory of integration, and also has had a serious interest in mathematical education. In 1964 he was awarded the Klement Gottwald State Prize, and in 1981 he was awarded the Bolzano medal of the Czechoslovak Academy of Sciences. Since 1989 he has been Director of the Mathematical Institute of the Czech Academy of Sciences in Prague and has had a profound influence on the mathematicians there. In 2006, he was awarded the ‘‘Czech Mind,’’ the highest scientific prize of the Czech Republic. Lebesgue integral—yet its scope is considerably greater; in particular, it includes functions that are derivatives, and also includes all ‘‘improper integrals.’’ In this chapter, we give an exposition of the generalized Riemann integral. In Section 10.1, it will be seen that the basic theory is almost exactly the same as for the ordinary Riemann integral. However, we have omitted the proofs of a few results when their proofs are unduly complicated. In the short Section 10.2, we indicate that improper integrals on [a, b] are included in the generalized theory. We will introduce the class of Lebesgue integrable functions as those generalized integrable functions f whose absolute value j f j is also generalized integrable; this is a very different approach to the Lebesgue integral than is usual, but it gives the same class of functions. In Section 10.3, we will integrate functions on unbounded closed intervals. In the final section, we discuss the limit theorems that hold for the generalized Riemann and Lebesgue integrals, and we will give some interesting applications of these theorems. We will also define what is meant by a ‘‘measurable function’’ and relate that notion to generalized integrability. Readers wishing to study the proofs that are omitted here should consult the first author’s book, A Modern Theory of Integration, which we refer to as [MTI], or the books of DePree and Swartz, Gordon, and McLeod listed in the References. Section 10.1 Definition and Main Properties In Definition 5.5.2, we defined a gauge on [a, b] to be a strictly positive function d : [a; b] ! ð0; 1Þ. Further, a tagged partition P_ :¼ fðI i ; ti Þgni¼1 of ½a; b, where I i :¼ [xi1 ; xi ], is said to be d-fine in the case ð1Þ ti 2 I i ½ti dðti Þ; ti þ dðti Þ for i ¼ 1; . . . ; n: C10 12/09/2010 15:0:8 290 Page 290 CHAPTER 10 THE GENERALIZED RIEMANN INTEGRAL This is shown in Figure 5.5.1. Note that (i) only a tagged partition can be d-fine, and (ii) the d-fineness of a tagged partition depends on the choice of the tags ti and the values dðti Þ. In Examples 5.5.4, we gave some specific examples of gauges, and in Theorem 5.5.5 we showed that if d is any gauge on [a, b], then there exist d-fine tagged partitions of [a, b]. We will define the generalized Riemann (or the ‘‘Henstock–Kurzweil’’) integral. It will be seen that the definition is very similar to that of the ordinary Riemann integral, and that many of the proofs are essentially the same. Indeed, the only difference between the definitions of these integrals is that the notion of smallness of a tagged partition is specified by a gauge, rather than its norm. It will be seen that this—apparently minor—difference results in a very much larger class of integrable functions. In order to avoid some complications, a few proofs will be omitted; they can be found in [MTI]. Before we begin our study, it is appropriate that we ask: Why are gauges more useful than norms? Briefly, the reason is that the norm of a partition is a rather coarse measure of the fineness of the partition, since it is merely the length of the largest subinterval in the partition. On the other hand, gauges can give one more delicate control of the subintervals in the partitions, by requiring the use of small subinterals when the function is varying rapidly but permitting the use of larger subintervals when the function is nearly constant. Moreover, gauges can be used to force specific points to be tags; this is often useful when unusual behavior takes place at such a point. Since gauges are more flexible than norms, their use permits a larger class of functions to become integrable. 10.1.1 Definition A function f : ½a; b ! R is said to be generalized Riemann integrable on [a, b] if there exists a number L 2 R such that for every e > 0 there exists a gauge de on [a, b] such that if P_ is any de -fine partition of [a, b], then jSð f ; PÞ Lj < e: The collection of all generalized Riemann integrable functions will usually be denoted by R ½a; b. It will be shown that if f 2 R ½a; b, then the number L is uniquely determined; it will be called the generalized Riemann integral of f over [a, b]. It will also be shown that if f 2 R½a; b, then f 2 R ½a; b and the value of the two integrals is the same. Therefore, it will not cause any ambiguity if we also denote the generalized Riemann integral of f 2 R ½a; b by the symbols Z b Z b f or f ðxÞ dx: a a Our first result gives the uniqueness of the value of the generalized Riemann integral. Although its proof is almost identical to that of Theorem 7.1.2, we will write it out to show how gauges are used instead of norms of partitions. 10.1.2 Uniqueness Theorem If f 2 R ½a; b, then the value of the integral is uniquely determined. Proof. Assume that L0 and L00 both satisfy the definition and let e > 0. Thus there exists a gauge d0e=2 such that if P_ 1 is any d0e=2 -fine partition, then jS f ; P 1 L0 j < e=2: C10 12/09/2010 15:0:9 Page 291 10.1 DEFINITION AND MAIN PROPERTIES 291 Also there exists a gauge d00e=2 such that if P_ 2 is any d00e=2 -fine partition, then jS f ; P 2 L00 j < e=2: n o We define de by de ðtÞ :¼ min d0e=2 ðtÞ; d00e=2 ðtÞ for t 2 ½a; b, so that de is a gauge on ½a; b. If P_ is a de -fine partition, then the partition P_ is both d0e=2 -fine and d00e=2 -fine, so that and jS f ; P L00 j < e=2; jS f ; P L0 j < e=2 whence it follows that jL0 L00 j jL0 S f ; P j þ jS f ; P L00 j < e=2 þ e=2 ¼ e: Since e > 0 is arbitrary, it follows that L0 ¼ L00 . Q.E.D. We now show that every Riemann integrable function f is also generalized Riemann integrable, and with the same value for the integral. This is done by using a gauge that is a constant function. 10.1.3 Consistency Theorem If f 2 R½a; b with integral L, then also f 2 R ½a; b with integral L. Proof. Given e > 0, we need to construct an appropriate gauge on [a, b]. Since f 2 R½a; b, there exists a number de > 0 such that if P_ is any tagged partition with P_ < de , then S f ; P_ L < e. We define the function d ðtÞ :¼ 1 de for t 2 ½a; b, so that e 4 de is a gauge on [a, b]. If P_ ¼ fðI i ; ti Þgni¼1 , where I i :¼ ½xi1 ; xi , is a de -fine partition, then since I i ti de ðti Þ; ti þ de ðti Þ ¼ ti 14 de ; ti þ 14 de ; it is readily seen that 0 < xi xi1 12 de < de for all i ¼ 1; . . . ; n. Therefore this partition _ L < e. also satisfies P_ < de and consequently Sð f : PÞ Thus every de -fine partition P_ also satisfies S f : P_ L < e. Since e > 0 is arbitrary, it follows that f is generalized Riemann integrable to L. Q.E.D. From Theorems 7.2.5, 7.2.7, and 7.2.8, we conclude that: Every step function, every continuous function, and every monotone function belongs to R ½a; b. We will now show that Dirichlet’s function, which was shown not to be Riemann integrable in 7.2.2(b) and 7.3.13(d), is generalized Riemann integrable. 10.1.4 Examples (a) The Dirichlet function f belongs to R ½0; 1 and has integral 0. We enumerate the rational numbers in [0, 1] as frk g1 k¼1 . Given e > 0 we define de ðrk Þ :¼ e=2kþ2 and de ðxÞ :¼ 1 when x is irrational. Thus de is a gauge on [0,1] and if the partition P_ :¼ fðI i ; ti Þgni¼1 is de -fine, then we have xi xi1 2de ðti Þ. Since the only nonzero contributions to S f ; P_ come from rational tags ti ¼ rk , where 0 < f ðrk Þðxi xi1 Þ ¼ 1 ðxi xi1 Þ 2e e ¼ ; 2kþ2 2kþ1 C10 12/09/2010 15:0:9 292 Page 292 CHAPTER 10 THE GENERALIZED RIEMANN INTEGRAL and since each such tag can occur in at most two subintervals, we have 1 1 X X 2e e 0 Sð f ; PÞ < ¼ ¼ e: kþ1 k 2 2 k¼1 k¼1 R1 Since e > 0 is arbitrary, then f 2 R ½0; 1 and 0 f ¼ 0. (b) Let H : ½0; 1 ! R be defined by Hð1=kÞ :¼ k for k 2 N and HðxÞ :¼ 0 elsewhere on [0, 1]. Since H is not bounded on [0, 1], it follows from the Boundedness Theorem 7.1.6 that it is not Riemann integrable on [0, 1]. We will now show that H is generalized Riemann integrable to 0. In fact, given e > 0, we define de ð1=kÞ :¼ e= k2kþ2 and set de ðxÞ :¼ 1 elsewhere on [0, 1], so de is a gauge on [0, 1]. If P_ is a de -fine partition of [0, 1] then xi xi1 2de ðti Þ. _ come from tags ti ¼ 1=k, where Since the only nonzero contributions to SðH; PÞ 0 < Hð1=kÞðxi xi1 Þ ¼ k ðxi xi1 Þ k 2e e ¼ kþ1 ; kþ2 k2 2 and since each such tag can occur in at most two subintervals, we have 1 X e 0 S H; P < ¼ e: k k¼1 2 Since e > 0 is arbitrary, then H 2 R ½0; 1 and R1 0 H ¼ 0: & The next result is exactly similar to Theorem 7.1.5. 10.1.5 Theorem Suppose that f and g are in R ½a; b. Then: (a) If k 2 R, the function kf is in R ½a; b and Z b Z kf ¼ k a (b) The function f þ g is in R ½a; b and Z b Z ð f þ gÞ ¼ a b a (c) If f ðxÞ gðxÞ for all x 2 ½a; b, then Z b Z f a a b a f: Z fþ b b a g: g: Proof. (b) Given e > 0, we can use the argument in the proof of the Uniqueness Theorem 10.1.2 to construct a gauge de on [a, b] such that if P_ is any de -fine partition of [a, b], then Z b Z b S f ; P f < e=2 and g < e=2: S g; P a a C10 12/09/2010 15:0:10 Page 293 10.1 DEFINITION AND MAIN PROPERTIES 293 Since S f þ g; P_ ¼ S f ; P_ þ S g; P_ , it follows as in the proof of Theorem 7.1.5(b) that Z b Z b Z b Z b S f þ g; P fþ g S f ; P f þ S g; P g a a a < e=2 þ e=2 ¼ e: a Since e > 0 is arbitrary, then f þ g 2 R ½a; b and its integral is the sum of the integrals of f and g. The proofs of (a) and (c) are analogous and are left to the reader. Q.E.D. It might be expected that an argument similar to that given in Theorem 7.1.6 can be used to show that a function in R ½a; b is necessarily bounded. However, that is not the case; indeed, we have already seen an unbounded function in R ½0; 1 in Example 10.1.4(b) and will encounter more later. However, it is a profitable exercise for the reader to determine exactly where the proof of Theorem 7.1.6 breaks down for a function in R ½a; b. The Cauchy Criterion There is an analogous form for the Cauchy Criterion for functions in R ½a; b. It is important because it eliminates the need to know the value of the integral. Its proof is essentially the same as that of 7.2.1. 10.1.6 Cauchy Criterion A function f : ½a; b ! R belongs to R ½a; b if and only if for _ are any partitions of every e > 0 there exists a gauge he on ½a, b such that if P_ and Q [a, b] that are he -fine, then S f ; P S f ; Q < e: _ Proof. ()) If f 2 R ½a; b with integral L, let de=2 be a gauge on [a, b] such that if P_ and Q are de=2 -fine partitions of [a, b], then jS f ; P Lj < e=2 and jS f ; Q Lj < e=2: _ are he -fine, then We set he ðtÞ :¼ de=2 ðtÞ for t 2 ½a; b, so if P_ and Q jS f ; P S f ; Q j jS f ; P Lj þ jL S f ; Q j < e=2 þ e=2 ¼ e: _ are partitions that (() For each n 2 N, let dn be a gauge on [a, b] such that if P_ and Q are dn -fine, then S f ; P S f ; Q < 1=n: We may assume that dn ðtÞ dnþ1 ðtÞ for all t 2 ½a; b and n 2 N; otherwise, we replace dn by the gauge d0n ðtÞ ¼ minfd1 ðtÞ; . . . ; dn ðtÞg for all t 2 ½a; b. For each n 2 N, let P_ n be a partition that is dn -fine. Clearly, if m > n then both P_ m and P_ n are dn -fine, so that S f ; P n S f ; P m < 1=n ð2Þ for m > n: 1 Consequently, the sequence S f ; P_ m m¼1 is a Cauchy sequence in R, so it converges to some number A. Passing to the limit in (2) as m ! 1, we have S f ; P n A 1=n for all n 2 N: C10 12/09/2010 15:0:11 294 Page 294 CHAPTER 10 THE GENERALIZED RIEMANN INTEGRAL To see that A is the generalized Riemann integral of f, given e > 0, let K 2 N satisfy _ is a dk -fine partition, then K > 2=e. If Q S f ; Q A S f ; Q S f ; P K þ S f ; P K A 1=K þ 1=K < e: Since e > 0 is arbitrary, then f 2 R ½a; b with integral A. Q.E.D. 10.1.7 Squeeze Theorem Let f : ½a; b ! R. Then f 2 R ½a; b if and only if for every e > 0 there exist functions ae and ve in R ½a; b with ae ðxÞ f ðxÞ ve ðxÞ and such that Z b a f or all x 2 ½a; b; ðve ae Þ e: The proof of this result is exactly similar to the proof of Theorem 7.2.3, and will be left to the reader. The Additivity Theorem We now present a result quite analogous to Theorem 7.2.9. Its proof is a modification of the proof of that theorem, but since it is somewhat technical, the reader may choose to omit the proof on a first reading. 10.1.8 Additivity Theorem Let f : ½a; b ! R and let c 2 ða; bÞ. Then f 2 R ½a; b if and only if its restrictions to ½a, c and ½c, b are both generalized Riemann integrable. In this case Z b Z c Z b ð3Þ f ¼ fþ f: a a c Proof. (() Suppose that the restriction f1 of f to [a, c] and the restriction f2 of f to [c, b] are generalized Riemann integrable to L1 and L2 , respectively. Then, given e > 0 thereexists a gauge d0 on [a, c] such that if P_ 1 is a d0 -fine partition of [a, c] then S f 1 ; P_ 1 L1 < e=2. 00 00 _ Also there exists a gauge d on [c, b] such that if P 2 is a d -fine partition of [c, b] then S f 2 ; P_ 2 L2 < e=2. We now define a gauge de on [a, b] by 8 0 1 for t 2 ½a; cÞ; > <min d ðtÞ; 2 ðc tÞ 0 00 for t ¼ c; de ðtÞ :¼ minfd ðcÞ; d ðcÞg > : 00 1 min d ðtÞ; 2 ðc tÞ for t 2 ðc; b: (This gauge has the property that any de -fine partition must have c as a tag for any subinterval containing the point c.) _ is any de -fine partition of [a, b], then there exist a d0 -fine We will show that if Q _ 1 of [a, c] and a d00 -fine partition Q _ 2 of [c, b] such that partition Q ð4Þ S f ; Q ¼ S f 1 ; Q1 þ S f 2 ; Q2 : _ then it belongs to two subintervals of Q _ and is the Case (i) If c is a partition point of Q, _ _ tag for both of these subintervals. If Q1 consists of the part of Q having subintervals in C10 12/09/2010 15:0:11 Page 295 10.1 DEFINITION AND MAIN PROPERTIES 295 _ 1 is d0 -fine. Similarly, if Q _ 2 consists of the part of Q _ having subintervals in [a, c], then Q 00 _ [c, b], then Q2 is d -fine. The relation (4) is now clear. _ ¼ fðI i ; ti Þgm , then it is the tag for some Case (ii) If c is not a partition point in Q i¼1 subinterval, say ½xk1 ; xk . We replace the pair ð½xk1 ; xk ; cÞ by the two pairs ð½xk1 ; c; cÞ _ 1 and Q _ 2 be the tagged partitions of [a, c] and [c, b] that result. and ð½c; xk ; cÞ, and let Q Since f ðcÞðxk xk1 Þ ¼ f ðcÞðc xk1 Þ þ f ðcÞðxk cÞ, it is seen that the relation (4) also holds. In either case, equation (4) and the Triangle Inequality imply that jSð f ; QÞ ðL1 þ L2 Þj ¼ jðSð f ; Q1 Þ þ Sð f ; Q2 ÞÞ ðL1 þ L2 Þj jSð f ; Q1 Þ L1 j þ jSð f ; Q2 Þ L2 j: _ 1 is d0 -fine and Q _ 2 is d00 -fine, we conclude that Since Q jSð f ; QÞ ðL1 þ L2 Þj < e: Since e > 0 is arbitrary, we infer that f 2 R ½a; b and that (3) holds. ()) Suppose that f 2 R ½a; b and, given e > 0, let the gauge he satisfy the Cauchy _ 1 be he -fine partitions of [a, c]. Criterion. Let f 1 be the restriction of f to [a, c] and let P_ 1 ; Q _ 1 to By adding additional partition points and tags from [c, b], we can extend P_ 1 and Q _ _ he -fine partitions P and Q of [a, b]. If we use the same additional points and tags in [c, b] _ then for both P_ and Q, Sð f ; PÞ Sð f ; QÞ ¼ Sð f 1 ; P 1 Þ Sð f 1 ; Q1 Þ: _ are he -fine, then jSð f ; P_ 1 Þ Sð f ; Q _ 1 Þj < e also holds. Therefore the Since both P_ and Q 1 1 Cauchy Condition shows that the restriction f 1 of f to [a, c] is in R ½a; c. Similarly, the restriction f 2 of f to [c, d] is in R ½c; d. The equality (3) now follows from the first part of the theorem. Q.E.D. It is easy to see that results exactly similar to 7.2.10–7.2.13 hold for the generalized Riemann integral. We leave their statements to the reader, but will use these results freely. The Fundamental Theorem (First Form) We will now give versions of the Fundamental Theorems for the generalized Riemann integral. It will be seen that the First Form is significantly stronger than for the (ordinary) Riemann integral; indeed, we will show that the derivative of any function automatically belongs to R ½a; b, so the integrability of the function becomes a conclusion, rather than a hypothesis. 10.1.9 The Fundamental Theorem of Calculus (First Form) Suppose there exists a countable set E in [a, b], and functions f ; F : ½a; b ! R such that: (a) F is continuous on [a, b]. (b) F 0 ðxÞ ¼ f ðxÞ f or all x 2 ½a; bnE: Then f belongs to R ½a; b and Z ð5Þ b a f ¼ FðbÞ FðaÞ: Proof. We will prove the theorem in the case where E ¼ ;, leaving the general case to be handled in the exercises. C10 12/09/2010 15:0:12 296 Page 296 CHAPTER 10 THE GENERALIZED RIEMANN INTEGRAL Thus, we assume that (b) holds for all x 2 ½a; b. Since we wish to show that f 2 R ½a; b, given e > 0, we need to construct a gauge de ; this will be done by using the differentiability of F on [a, b]. If t 2 I, since the derivative f ðtÞ ¼ F 0 ðtÞ exists, there exists de ðtÞ > 0 such that if 0 < jz tj de ðtÞ; z 2 ½a; b, then FðzÞ FðtÞ 1 < e: f ðtÞ zt 2 If we multiply this inequality by jz tj, we obtain jFðzÞ FðtÞ f ðtÞðz tÞj 12 ejz tj whenever z 2 ½t de ðtÞ; t þ de ðtÞ \ ½a; b. The function de is our desired gauge. Now let u; v 2 ½a; b with u < v satisfy t 2 ½u; v ½t de ðtÞ; t þ de ðtÞ. If we subtract and add the term FðtÞ f ðtÞ t and use the Triangle Inequality and the fact that v t 0 and t u 0, we get jFðvÞ FðuÞ f ðtÞðv uÞj jFðvÞ FðtÞ f ðtÞðv tÞj þ jFðtÞ FðuÞ f ðtÞðt uÞj 12 eðv tÞ þ 12 eðt uÞ ¼ 12 eðv uÞ: Therefore, if t 2 ½u; v ½t de ðtÞ; t þ de ðtÞ, then we have ð6Þ jFðvÞ FðuÞ f ðtÞðv uÞj 12 eðv uÞ: We will show that f 2 R ½a; b with integral given by the telescoping sum n X fFðxi Þ Fðxi1 Þg: ð7Þ FðbÞ FðaÞ ¼ i¼1 For, if the partition P_ :¼ fð½xi1 ; xi ; ti Þgni¼1 is de -fine, then ti 2 ½xi1 ; xi ½ti de ðti Þ; ti þ de ðti Þ for i ¼ 1; . . . ; n; and so we can use (7), the Triangle Inequality, and (6) to obtain X n jFðbÞ FðaÞ Sð f ; PÞj¼ fFðxi Þ Fðxi1 Þ f ðti Þðxi xi1 Þg i¼1 n X jFðxi Þ Fðxi1 Þ f ðti Þðxi xi1 Þj i¼1 n X i¼1 1 2 eðxi xi1 Þ < eðb aÞ: Since e > 0 is arbitrary, we conclude that f 2 R ½a; b and (5) holds. Q.E.D. pffiffiffi 10.1.10 Examples (a) If HðxÞ :¼ 2 x for x 2 ½0; b, then H is continuous on ½0; b and p ffiffiffi H 0 ðxÞ ¼ 1= x for x 2 ð0; b. We define hðxÞ :¼ H 0 ðxÞ for x 2 ð0; b and hð0Þ :¼ 0. It follows from R b the Fundamental Theorem 10.1.9 with E :¼ f0g that h belongs to R ½0; b and that 0 h ¼ HðbÞ Hð0Þ ¼ HðbÞ, which we write as Z b pffiffiffi 1 pffiffiffi dx ¼ 2 b: x 0 (b) More generally, if a > 0, let H a ðxÞ :¼ xa =a ¼ ea ln x =a for x 2 ð0; b and H a ð0Þ :¼ 0 so that H a is continuous on [0, b] and H 0a ðxÞ ¼ xa1 for all x 2 ð0; b; see 8.3.10 and 8.3.13. We define ha ðxÞ :¼ Ha0 ðxÞ for x 2 ð0; b and ha ð0Þ :¼ 0. C10 12/09/2010 15:0:12 Page 297 10.1 DEFINITION AND MAIN PROPERTIES Then Theorem 10.1.9 implies that ha 2 R ½0; b and that H a ðbÞ, which we write as Z b ba xa1 dx ¼ : a 0 Rb 0 297 ha ¼ H a ðbÞ H a ð0Þ ¼ (c) Let LðxÞ :¼ x ln x x for x 2 ð0; b and Lð0Þ :¼ 0. Then L is continuous on [0, b] (use l’Hospital’s Rule at x ¼ 0), and it is seen that L0 ðxÞ ¼ ln x for x 2 ð0; b. It follows from Theorem 10.1.9 with E ¼ f0g thatR the unbounded function lðxÞ :¼ ln x b for x 2 ð0; b and lð0Þ :¼ 0 belongs to R ½0; b and that 0 l ¼ LðbÞ Lð0Þ, which we write as Z b ln x dx ¼ b ln b b: 0 (d)pffiffiffiffiffiffiffiffiffiffiffiffiffi Let AðxÞ :¼ Arcsin x for x 2 ½1; 1 so that A is continuous on ½1; 1 and A0 ðxÞ ¼ 1= 1 x2 for x 2 ð1; 1Þ. We define sðxÞ :¼ A0 ðxÞ for x 2 ð1; 1Þ and let sð1Þ ¼ sð1Þ :¼ 0. R 1 Then Theorem 10.1.9 with E ¼ f1; 1g implies that s 2 R ½1; 1 and that 1 s ¼ Að1Þ Að1Þ ¼ p, which we write as Z 1 dx & pffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ Arcsin 1 Arcsinð1Þ ¼ p: 1 1 x2 The Fundamental Theorem (Second Form) We now turn to the Second Form of the Fundamental Theorem, in which we wish to differentiate the indefinite integral F of f, defined by: Z z ð8Þ FðzÞ :¼ f ðxÞ dx for z 2 ½a; b: a 10.1.11 Fundamental Theorem of Calculus (Second Form) Let f belong to R ½a; b and let F be the indefinite integral of f. Then we have: (a) F is continuous on [a, b]. (b) There exists a null set Z such that if x 2 ½a; bnZ, then F is differentiable at x and F 0 ðxÞ ¼ f ðxÞ. (c) If f is continuous at c 2 ½a; b, then F 0 ðcÞ ¼ f ðcÞ. Proof. The proofs of (a) and (b) can be found in [MTI]. The proof of (c) is exactly as the proof of Theorem 7.3.5 except that we use Theorems 10.1.5(c) and 10.1.8. Q.E.D. We can restate conclusion (b) as: The indefinite integral F of f is differentiable to f almost everywhere on [a, b]. Substitution Theorem In view of the simplicity of the Fundamental Theorem 10.1.9, we can improve the theorem justifying the ‘‘substitution formula.’’ The next result is a considerable strengthening of Theorem 7.3.8. The reader should write out the hypotheses in the case Ef ¼ Ew ¼ E ¼ ;. 10.1.12 Substitution Theorem (a) Let I :¼ ½a; b and J :¼ ½a; b, and let F : I ! R and w : J ! R be continuous functions with wðJÞ I. C10 12/09/2010 15:0:13 298 Page 298 CHAPTER 10 THE GENERALIZED RIEMANN INTEGRAL (b) Suppose there exist sets Ef I and Ew J such that f ðxÞ ¼ F 0 ðxÞ for x 2 InEf , that w0 ðtÞ exists for t 2 JnEw , and that E :¼ w1 ðEf Þ [ Ew is countable. (c) Set f ðxÞ :¼ 0 for x 2 Ef and w0 ðtÞ :¼ 0 for t 2 Ew . We conclude that f 2 R ðwðJÞÞ, that ð f wÞ w0 2 R ðJÞ and that Z b b Z wðbÞ 0 ð f wÞ w ¼ F w ¼ f: ð9Þ a a wðaÞ Proof. Since w is continuous on J, Theorem 5.3.9 implies that wðJÞ is a closed interval in I. Also w1 ðEf Þ is countable, whence Ef \ wðJÞ ¼ wðw1 ðEf ÞÞ is also countable. Since f ðxÞ ¼ F 0 ðxÞ for all x 2 wðJÞnEf , the Fundamental Theorem 10.1.9 implies that f 2 R ðwðJÞÞ and that Z wðbÞ wðbÞ f ¼ F ¼ FðwðbÞÞ FðwðaÞÞ: wðaÞ wðaÞ If t 2 JnE, then t 2 JnEw and wðtÞ 2 InEf . Hence the Chain Rule 6.1.6 implies that ðF wÞ0 ðtÞ ¼ f ðwðtÞÞ w0 ðtÞ for t 2 JnE:: Since E is countable, the Fundamental Theorem implies that ð f wÞ w0 2 R ðJÞ and that Z b b ð f wÞ w0 ¼ F w ¼ FðaðbÞÞ FðwðaÞÞ: a a The conclusion follows by equating these two terms. Q.E.D. pffiffi cos t pffiffi dt. 10.1.13 Examples (a) Consider the integral t 0 Since the integrand is unbounded as t ! 0þ, there is some doubt about the existence of the integral.pAlso, we have seen in Exercise 7.3.19(b) that Theorem 7.3.8 does not apply ffiffi with wðtÞ :¼ t. However, Theorem 10.1.12 applies. pffiffi Indeed, this substitution gives w0 ðtÞ ¼ 1=ð2 tÞ for t 2 ð0; 4 and we set w0 ð0Þ :¼ 0. If we put FðxÞ :¼ 2 sin x, then f ðxÞ ¼ F 0 ðxÞ ¼ 2 cos x and the integrand has the form pffiffi 1 pffiffi f ðwðtÞÞ w0 ðtÞ ¼ 2 cos t for t 6¼ 0: 2 t Z 4 Thus, the Substitution Theorem 10.1.12 with Ew :¼ f0g; Ef :¼ ;; E :¼ f0g implies that pffiffi Z t¼4 Z x¼2 cos t pffiffi dt ¼ 2 cos x dx ¼ 2 sin 2: t t¼0 x¼0 Z Z 1 dt dt pffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffipffiffiffiffiffiffiffiffiffiffi : 2 t 1t tt 0 0 Note that this integrand is unbounded as t ! 0þ and as t ! 1. As in (a), we let pffiffiffiffiffiffiffiffiffiffi pffiffi pffiffi x ¼ wðtÞ :¼ t for t 2 ½0; 1 so that w0 ðtÞ ¼ 1= 2 t for t 2 ð0; 1. Since 1 t ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 , the integrand takes the form 1 (b) Consider the integral 2 1 2 pffiffiffiffiffiffiffiffiffiffi pffiffi ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi w0 ðtÞ; 2 t 1t 1 x2 C10 12/09/2010 15:0:13 Page 299 10.1 DEFINITION AND MAIN PROPERTIES 299 pffiffiffiffiffiffiffiffiffiffiffiffiffi which suggests f ðxÞ ¼ 2= 1 x2 for x 6¼ 1. Therefore, we are led to choose FðxÞ :¼ 2 Arcsin x for x 2 ½0; 1, since 2 pffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ F 0 ðxÞ ¼ ð2 Arcsin xÞ0 for x 2 ½0; 1Þ: 1 x2 Consequently, we have Ew ¼ f0g and Ef ¼ f1g, so that E ¼ f0; 1g, and the Substitution Theorem yields Z t¼1 Z x¼1 1 dt 2dx & pffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2 Arcsin x ¼ 2 Arcsin 1 ¼ p: pffiffipffiffiffiffiffiffiffiffiffiffi ¼ 2 0 t 1 t 1x t¼0 x¼0 Other formulations of the Substitution Theorem are given in [MTI]. The Multiplication Theorem In Theorem 7.3.16 we saw that the product of two Riemann integrable functions is Riemann integrable. That result is not true for generalized Riemann integrable functions; see Exercises 18 and 20. However, we will state a theorem in this direction that is often useful. Its proof will be found in [MTI]. 10.1.14 Multiplication Theorem If f 2 R ½a; b and if g is a monotone function on [a, b], then the product f g belongs to R ½a; b. Integration by Parts The following version of the formula for integration by parts is useful. 10.1.15 Integration by Parts Theorem Let F and G be differentiable on [a, b]. Then F0 G belongs to R ½a; b if and only if FG0 belongs to R ½a; b. In this case we have b Z b Z b F 0 G ¼ FG FG0 : ð10Þ a a a The proof uses Theorem 6.1.3(c); it will be left to the reader. In applications, we usually have F 0 ðxÞ ¼ f ðxÞ and G0 ðxÞ ¼ gðxÞ for all x 2 ½a; b. It will be noted that we need to assume that one of the functions f G ¼ F 0 G and Fg ¼ FG0 belongs to R ½a; b. The reader should contrast the next result with Theorem 7.3.18. Note that we do not need to assume the integrability of f ðnþ1Þ . 10.1.16 Taylor’s Theorem Suppose that f ; f 0 ; f 00 ; . . . ; f ðnÞ and f ðnþ1Þ exist on [a, b]. Then we have ð11Þ f ðbÞ ¼ f ðaÞ þ f 0 ðaÞ f ðnÞ ðaÞ ðb aÞ þ þ ðb aÞn þ Rn ; 1! n! where the remainder is given by ð12Þ Rn ¼ 1 n! Z a b f ðnþ1Þ ðtÞ ðb tÞn dt: Proof. Since f ðnþ1Þ is a derivative, it belongs to R ½a; b. Moreover, since t 7! ðb tÞn is monotone on [a, b], the Multiplication Theorem 10.1.14 implies the integral in (12) exists. Integrating by parts repeatedly, we obtain (11). Q.E.D. C10 12/09/2010 15:0:14 300 Page 300 CHAPTER 10 THE GENERALIZED RIEMANN INTEGRAL Exercises for Section 10.1 1. Let (a) (b) (c) (d) 2. (a) (b) 3. Let (a) (b) (c) d be a gauge on [a, b] and let P_ ¼ fð½xi1 ; xi ; ti Þgni¼1 be a d-fine partition of [a, b]. Show that 0 < xi xi1 2d(ti) for i ¼ 1; . . . ; n. _ 2d . If d :¼ supfdðtÞ : t 2 ½a; bg < 1, show that kPk _ is a tagged partition of [a, b] such If d :¼ inffdðtÞ : t 2 ½a; bg satisfies d > 0, and if Q _ d , show that Q _ is d-fine. that we have kQk If e ¼ 1, show that the gauge d1 in Example 10.1.4(a) has the property that inffd1 ðtÞ : t 2 ½0; 1g ¼ 0. If P_ is a tagged partition of [a, b], show that each tag can belong to at most two subintervals _ in P. Are there tagged partitions in which every tag belongs to exactly two subintervals? d be a gauge on [a, b] and let P_ be a d-fine partition of [a, b]. _ 1 such that (i) no tag belongs to two subintervals Show that there exists a d-fine partition Q _ _ _ in Q1 , and (ii) Sð f ; Q1 Þ ¼ Sð f ; PÞ for any function f on [a, b]. _ 2 such that (j) every tag belongs to two subintervals in Does there exist a d-fine partition Q _ 2 Þ ¼ Sð f ; PÞ _ 2 , and (jj) Sð f ; Q _ for any function f on [a, b]? Q _ 3 such that (k) every tag is an endpoint of its Show that there exists a d-fine partition Q _ _ subinterval, and (kk) Sð f ; Q3 Þ ¼ Sð f ; PÞ for any function f on [a, b]. 4. If d is defined on [0, 2] by dðtÞ :¼ 12 jt 1j for x 6¼ 1 and dð1Þ :¼ 0:01, show that every d-fine partition P_ of [0, 2] has t ¼ 1 as a tag for at least one subinterval, and that the total length of the subintervals in P_ having 1 as a tag is 0:02. 5. (a) Construct a gauge d on [0, 4] that will force the numbers 1, 2, 3 to be tags of any d-fine partition of this interval. (b) Given a gauge d1 on [0, 4], construct a gauge d2 such that every d2 -fine partition of [0, 4] will (i) have the numbers 1, 2, 3 in its collection of tags, and (ii) be d1 -fine. 6. Show that f 2 R ½a; b with integral L if and only if for every e > 0 there exists a gauge g e on [a, b] such that if P_ ¼ fð½xi1 ; xi ; ti Þgni¼1 is any tagged partition such that 0 < xi xi1 _ Lj < e. (This provides an alternate—but equivalent— g e ðti Þ for i ¼ 1; . . . ; n, then jSð f ; PÞ way of defining the generalized Riemann integral.) 7. Show that the following functions belong to R ½0; 1 by finding a function F k that is continuous on [0, 1] and such that F 0k ðxÞ ¼ f k ðxÞ for x 2 ½0; 1nEk , for some finite set Ek. pffiffiffi (a) f 1 ðxÞ :¼ ðx þ 1Þ= x for x 2 ð0; 1 and f 1 ð0Þ :¼ 0. pffiffiffiffiffiffiffiffiffiffiffi (b) f 2 ðxÞ :¼ x= 1 x for x 2 ½0; 1Þ and f 2 ð1Þ :¼ 0. pffiffiffi (c) f 3 ðxÞ :¼ x ln x for x 2 ð0; 1 and f 3 ð0Þ :¼ 0. pffiffiffi (d) f 4 ðxÞ :¼ ðln xÞ= x for x 2 ð0; 1 and f 4 ð0Þ :¼ 0. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (e) f 5 ðxÞ :¼ ð1 þ xÞ=ð1 xÞ for x 2 ½0; 1Þ and f 5 ð1Þ :¼ 0. pffiffiffipffiffiffiffiffiffiffiffiffiffiffi (f) f 6 ðxÞ :¼ 1=ð x 2 xÞ for x 2 ð0; 1 and f 6 ð0Þ :¼ 0. 8. Explain why the argument in Theorem 7.1.6 does not apply to show that a function in R ½a; b is bounded. 9. Let f ðxÞ :¼ 1=x for x 2 ð0; 1 and f ð0Þ :¼ 0; then f is continuous except at x ¼ 0. Show that f does not belong to R ½0; 1. [Hint: Compare f with sn ðxÞ :¼ 1 on ð1=2; 1; sn ðxÞ :¼ 2 on ð1=3; 1=2; sn ðxÞ :¼ 3 on ð1=4; 1=3; . . . ; sn ðxÞ :¼ n on [0, 1=n].] 10. Let k : ½0; 1 ! R be defined by kðxÞ :¼ 0 if x 2 ½0; 1 is 0 or is irrational, and kðm=nÞ :¼ n if m; n 2 N have no common integer factors other than 1. Show that k 2 R ½0; 1 with integral equal to 0. Also show that k is not continuous at any point, and not bounded on any subinterval [c, d] with c < d. 11. Let f be Dirichlet’s function on [0, 1] and FðxÞ :¼ 0 for all x 2 ½0; 1. Since F 0 ðxÞ ¼ f ðxÞ for all x 2 ½0; 1nQ , show that the Fundamental Theorem 10.1.9 implies that f 2 R ½0; 1. C10 12/09/2010 15:0:16 Page 301 10.1 DEFINITION AND MAIN PROPERTIES 301 12. Let MðxÞ :¼ lnjxj for x 6¼ 0 and Mð0Þ :¼ 0. Show that M 0 ðxÞ ¼ 1=x for all x 6¼ 0. Explain why R2 it does not follow that 2 ð1=xÞdx ¼ lnj2j ln 2 ¼ 0. 13. Let L1 ðxÞ :¼ x ln jxj x for x 6¼ 0 and L1 ð0Þ :¼ 0, and let l 1 ðxÞ :¼ lnjxj if x 6¼ 0 and Rb l 1 ð0Þ :¼ 0. If [a, b] is any interval, show that l 1 2 R ½a; b and that a lnjxjdx ¼ L1 ðbÞ L1 ðaÞ. 14. Let E :¼ fc1 ; c2 ; . . .g and let F be continuous on [a, b] and F 0 ðxÞ ¼ f ðxÞ for x 2 ½a; bnE and f ðck Þ :¼ 0. We want to show that f 2 R ½a; b and that equation (5) holds. (a) Given e > 0 and t 2 ½a; bnE, let de ðtÞ be defined as in the proof of 10.1.9. Choose de ðck Þ > 0 such that if jz ck j < de ðck Þ and z 2 ½a; b, then jFðzÞ Fðck Þj < e=2kþ2 . (b) Show that if the partition P_ is de -fine and has a tag ti ¼ ck , then we have jFðxi Þ Fðxi1 Þ f ðck Þðxi xi1 Þj < e=2kþ1 . _ ðFðbÞ FðaÞÞj < eðb a þ 1Þ. (c) Use the argument in 10.1.9 to get jSð f ; PÞ 15. Show that the function g1 ðxÞ :¼ x1=2 sinð1=xÞ for x 2 ð0; 1 and g1 ð0Þ :¼ 0 belongs to R ½0; 1. [Hint: Differentiate C1 ðxÞ :¼ x3=2 cosð1=xÞ for x 2 ð0; 1 and C1 ð0Þ :¼ 0.] 16. Show that the function g2 ðxÞ :¼ ð1=xÞsinð1=xÞ for x 2 ð0; 1 and g2 ð0Þ :¼ 0 belongs to R ½0; 1. [Hint: Differentiate C2 ðxÞ :¼ x cosð1=xÞ for x 2 ð0; 1 and C2 ð0Þ :¼ 0, and use the result for the cosine function that corresponds to Exercise 7.2.12.] 17. Use the Substitution Theorem 10.1.12 to evaluate the following integrals: Z 4 pffiffi Z 3 t dt 2 pffiffi ; ð2t þ 1Þsgnðt þ t 2Þdt ¼ 6; (b) (a) 1 þ t 3 0 Z 1 pffiffiffiffiffiffiffiffiffiffiffiffi Z 5 dt pffiffiffiffiffiffiffiffiffiffi ¼ 2 Arctan 2; 1 t2 dt: (d) (c) 1 t t1 0 18. Give an example of a function f 2 R ½0; 1 whose square f 2 does not belong to R ½0; 1. 19. Let FðxÞ :¼ x cosðp=xÞ for x 2 ð0; 1 and Fð0Þ :¼ 0. It will be seen that f :¼ F 0 2 R ½0; 1 but that its absolute value j f j ¼ jF 0 j 2 = R ½0; 1. (Here f ð0Þ :¼ 0.) (a) Show that F 0 and jF 0 j are continuous on any interval ½c; 1; 0 < c < 1 and f 2 R ½0; 1. (b) If ak :¼ 2=ð2k þ 1Þ and bk :¼ 1=k for k 2 N, then the intervals ½ak ; bk are nonRb overlapping and 1=k akk j f j. 1 X 1=k diverges, then j f j 2 = R ½0; 1. (c) Since the series k¼1 20. Let f be as in Exercise 19 and let mðxÞ :¼ ð1Þk for x 2 ½ak ; bk ðk 2 NÞ, and mðxÞ :¼ 0 elsewhere in [0, 1]. Show that m f ¼ jm f j. Use Exercise 7.2.11 to show that the bounded functions m and jmj belong to R½0; 1. Conclude that the product of a function in R ½0; 1 and a bounded function in R½0; 1 may not belong to R ½0; 1. 21. Let FðxÞ :¼ xjcosðp=xÞj for x 2 ð0; 1 and let Fð0Þ :¼ 0. Then F is continuous on [0, 1] and F0 ðxÞ exists for x 2 = E :¼ f0g [ fak : k 2 Ng, where ak :¼ 2=ð2k þ 1Þ. Let wðxÞ :¼ F0 ðxÞ for x2 = E and wðxÞ :¼ 0 for x 2 E. Show that w is not bounded on [0,1].RUsing the Fundamental b Theorem 10.1.9 with E countable, conclude that w 2 R ½0; 1 and that a w ¼ FðbÞ FðaÞ for a; b 2 ½0; 1. As in Exercise 19, show that jwj 2 = R ½0; 1. 22. Let CðxÞ :¼ x2 jcosðp=xÞj for x 2 ð0; 1 and Cð0Þ :¼ 0. Then C is continuous on [0, 1] and C0 ðxÞ exists for x 2 = E1 :¼ fak g. Let CðxÞ :¼ C0 ðxÞ for x 2 = E1, and cðxÞ :¼ 0 for x 2 E1. Show that c is bounded on [0, 1] and (using Exercise 7.2.11) that c 2 R½0; 1. Show that Rb a c ¼ CðbÞ CðaÞ for a; b 2 ½0; 1. Also show that jcj 2 R½0; 1. 23. If f : ½a; b ! R is continuous and if p 2 R ½a; b does not change sign on [a, b], and if Rb Rb f p 2 R ½a; b, then there exists j 2 ½a; b such that a f p ¼ f ðjÞ a p. (This is a generalization of Exercise 7.2.16; it is called the First Mean Value Theorem for integrals.) 24. Let f 2 R ½a; b, let g be monotone on [a, b] and suppose that f 0. Then there exists j 2 ½a; b Rb Rj Rb such that a f g ¼ gðaÞ a f þ gðbÞ j f . (This is a form of the Second Mean Value Theorem for integrals.) C10 12/09/2010 15:0:16 302 Page 302 CHAPTER 10 THE GENERALIZED RIEMANN INTEGRAL Section 10.2 Improper and Lebesgue Integrals We have seen in Theorem 7.1.6 that a function f in R½a; b must be bounded on [a, b] (although this need not be the case for a function in R ½a; b). In order to integrate certain functions that have infinite limits at a point c in [a, b], or that are highly oscillatory at such a point, one learns in calculus to take limits of integrals over subintervals, as the endpoints of these subintervals tend to the point c. pffiffiffi For example, the function hðxÞ :¼ 1= x for x 2 ð0; 1 and hð0Þ :¼ 0 is unbounded on a neighborhood of the left endpoint of [0, 1]. However, it does belong to R½g; 1 for every g 2 ð0; 1 and we define the ‘‘improper Riemann integral’’ of h on [0, 1] to be the limit Z 1 Z 1 1 1 pffiffiffi dx :¼ lim pffiffiffi dx: g!0þ x x 0 g We would treat the oscillatory function kðxÞ :¼ sinð1=xÞ for x 2 ð0; 1 and kð0Þ :¼ 0 in the same way. One handles a function that becomes unbounded, or is highly oscillatory, at the right endpoint of the interval in a similar fashion. Furthermore, if a function g is unbounded, or is highly oscillatory, near some c 2 ða; bÞ, then we define the ‘‘improper Riemann integral’’ to be Z b Z a Z b ð1Þ g :¼ lim g þ lim g: a a!c a b!cþ b These limiting processes are not necessary when one deals with the generalized Riemann integral. pffiffiffi For example,pwe have seen in Example 10.1.10(a) that if HðxÞ :¼ 2 x for x 2 ½0; 1 ffiffiffi then H 0 ðxÞ ¼ 1= x ¼: hðxÞ for x 2 ð0; 1 and the Fundamental Theorem 10.1.9 asserts that h 2 R ½0; 1 and that Z 0 1 1 pffiffiffi dx ¼ Hð1Þ Hð0Þ ¼ 2: x This example is an instance of a remarkable theorem due to Heinrich Hake, which we now state in the case where the function becomes unbounded or is oscillatory near the right endpoint of the interval. 10.2.1 Hake’s Theorem If f : ½a; b ! R, then f 2 R ½a; b if and only if for every g 2 ða; bÞ the restriction of f to ½a; g belongs to R ½a; g and Z g f ¼ A 2 R: ð2Þ lim g!b Z In this case b a a f ¼ A. The idea of the proof of the ð(Þ part of this resultRis to take an increasing sequence g ðg n Þ converging to b so that f 2 R ½a; g n and limn a n f ¼ A. In order to show that f 2 R ½a; b, we need to construct gauges on [a, b]. This is done by carefully ‘‘piecing together’’ gauges that work for the intervals ½g i1 ; g i to obtain a gauge on [a, b]. Since the details of this construction are somewhat delicate and not particularly informative, we will not go through them here but refer the reader to [MTI]. C10 12/09/2010 15:0:17 Page 303 10.2 IMPROPER AND LEBESGUE INTEGRALS 303 It is important to understand the significance of Hake’s Theorem. It implies that the generalized Riemann integral cannot be extended by taking limits as in (2). Indeed, if a function f has the property that its restriction to every subinterval [a, g], where g 2 ða; bÞ, is generalized Riemann integrable and such that (2) holds, then f already belongs to R ½a; b. An alternative way of expressing this fact is that the generalized Riemann integral does not need to be extended by taking such limits. One can test a function for integrability on [a, b] by examining its behavior on subintervals [a, g] with g < b. Since it is usually difficult to establish that a function is in R ½a; b by using Definition 10.1.1, this fact gives us another tool for showing that a function is generalized Riemann integrable on [a, b]. It is often useful to evaluate the integral of a function by using (2). We will use these observations to give an important example that provides insight into the set of generalized Riemann integrable functions. 1 X 10.2.2 Example (a) Let ak be any series of real numbers converging to A 2 R. We k¼1 will construct a function w 2 R ½0; 1 such that Z 1 1 X w¼ ak ¼ A: 0 k¼1 Indeed, we define w : ½0; 1! R to be the function that takes the values 2a1 ; 22 a2 ; 23 a3 ; . . . on the intervals 0; 12Þ; 12 ; 34Þ; 34 ; 78Þ; . . .. (See Figure 10.2.1.) For convenience, let ck :¼ 1 1=2k for k ¼ 0; 1; . . . ; then wðxÞ :¼ 2k a k 0 ck1 x < ck ðk 2 NÞ; x ¼ 1: for for Figure 10.2.1 The graph of w Clearly the restriction of w to each interval ½0; g for g 2 ð0; 1Þ, is a step function and therefore is integrable. In fact, if g 2 ½cn ; cnþ1 Þ then Z g 1 1 1 2 n w ¼ ð2a1 Þ þ ð2 a2 Þ 2 þ þ ð2 an Þ n þ rg 2 2 2 0 ¼ a1 þ a2 þ þ an þ r g ; where jrg j janþ1 j. But since the series is convergent, then rg ! 0 and so Z g n X lim w ¼ lim ak ¼ A: g!1 0 n!1 k¼1 C10 12/09/2010 15:0:18 304 Page 304 CHAPTER 10 THE GENERALIZED RIEMANN INTEGRAL (b) If the series 1 X ak is absolutely convergent in the sense of Definition 9.1.1, then it k¼1 follows as in (a) that the function jwj also belongs to R ½0; 1 and that Z 1 1 X jwj ¼ jak j: 0 k¼1 1 X jak j is not convergent, then the function jwj does not belong to However, if the series k¼1 R ½0; 1. Since there are many convergent series that are not absolutely convergent (for 1 X example, ð1Þk =k), we have examples of functions that belong to R ½0; 1 but whose k¼1 absolute values do not belong to R ½0; 1. We have already encountered such functions in & Exercises 10.1.19 and 10.1.21. The fact that there are generalized Riemann integrable functions whose absolute value is not generalized Riemann integrable is often summarized by saying that the generalized Riemann integral is not an ‘‘absolute integral.’’ Thus, in passing to the generalized Riemann integral we lose an important property of the (ordinary) Riemann integral. But that is the price that one must pay in order to be able to integrate a much larger class of functions. Lebesgue Integrable Functions In view of the importance of the subset of functions in R ½a; b whose absolute values also belong to R ½a; b, we will introduce the following definition. 10.2.3 Definition A function f 2 R ½a; b such that j f j 2 R ½a; b is said to be Lebesgue integrable on [a, b]. The collection of all Lebesgue integrable functions on [a, b] is denoted by L½a; b. Note The collection of all Lebesgue integrable functions is usually introduced in a totally different manner. One of the advantages of the generalized Riemann integral is that it includes the collection of Lebesgue integrable functions as a special—and easily identifiable—collection of functions. It is clear that if f 2 R ½a; b and if f ðxÞ 0 for all x 2 ½a; b, then we have j f j ¼ f 2 R ½a; b, so that f 2 L½a; b. That is, a nonnegative function f 2 R ½a; b belongs to L½a; b. The next result gives a more powerful test for a function in R ½a; b to belong to L½a; b. 10.2.4 Comparison Test If f ; v 2 R ½a; b and j f ðxÞj vðxÞ for all x 2 ½a; b, then f 2 L½a; b and Z b Z b Z b f j f j v: ð3Þ a a a Partial Proof. The fact that j f j 2 R ½a; b is proved in [MTI]. Since j f j 0, this implies that f 2 L½a; b. To establish (3), we note that j f j f j f j and 10.1.5(c) imply that Z b Z b Z b jfj f j f j; a a a C10 12/09/2010 15:0:18 Page 305 10.2 IMPROPER AND LEBESGUE INTEGRALS 305 whence the first inequality in (3) follows. The second inequality follows from another application of 10.1.5 (c). Q.E.D. The next result shows that constant multiples and sums of functions in L½a; b also belong to L½a; b. 10.2.5 Theorem If f ; g 2 L½a; b and if c 2 R, then cf and f þ g also belong to L½a; b. Moreover Z ð4Þ b a Z cf ¼ c a b Z f and b a Z j f þ gj b a Z jfj þ b a jgj: Proof. Since jcf ðxÞj ¼ jcjj f ðxÞj for all x 2 ½a; b, the hypothesis that j f j belongs to R ½a; b implies that cf and jcf j also belong to R ½a; b, whence cf 2 L½a; b. The Triangle Inequality implies that j f ðxÞ þ gðxÞj j f ðxÞj þ jgðxÞj for all x 2 ½a; b. But since v :¼ j f j þ jgj belongs to R ½a; b, the Comparison Test 10.2.4 implies that f þ g belongs to L½a; b and that Z b a Z j f þ gj a b Z ðj f j þ jgjÞ ¼ b a Z jfj þ a b jgj: Q.E.D. The next result asserts that one only needs to establish a one-sided inequality in order to show that a function f 2 R ½a; b actually belongs to L½a; b. 10.2.6 Theorem If f 2 R ½a; b, the following assertions are equivalent: (a) f 2 L½a; b. (b) There exists v 2 L½a; b such that f ðxÞ vðxÞ for all x 2 ½a; b. (c) There exists a 2 L½a; b such that aðxÞ f ðxÞ for all x 2 ½a; b. Proof. (a) ) (b) Let v :¼ f . (b) ) (a) Note that f ¼ v ðv f Þ. Since v f 0 and since v f belongs to R ½a; b, it follows that v f 2 L½a; b. Now apply Theorem 10.2.5. We leave the proof that (a) () (c) to the reader. 10.2.7 Theorem belong to L½a; b. Q.E.D. If f ; g 2 L½a; b, then the functions maxf f ; gg and minf f ; gg also Proof. It follows from Exercise 2.2.18 that if x 2 ½a; b, then maxf f ðxÞ; gðxÞg ¼ 12 ð f ðxÞ þ gðxÞ þ j f ðxÞ gðxÞjÞ; minf f ðxÞ; gðxÞg ¼ 12 ð f ðxÞ þ gðxÞ j f ðxÞ gðxÞjÞ: The assertions follow from these equations and Theorem 10.2.5. Q.E.D. In fact, the preceding result gives a useful conclusion about the maximum and the minimum of two functions in R ½a; b. C10 12/09/2010 15:0:19 306 Page 306 CHAPTER 10 THE GENERALIZED RIEMANN INTEGRAL 10.2.8 Theorem Suppose that f ; g; a, and v belong to R ½a; b. If f v; g v or if a f ; a g; then maxf f ; gg and minf f ; gg also belong to R ½a; b. Proof. Suppose that f v and g v; then maxf f ; gg v. It follows from the first equality in the proof of Theorem 10.2.7 that 0 j f gj ¼ 2 maxf f ; gg f g 2v f g: Since 2v f g 0, this function belongs to L½a; b. The Comparison Test 10.2.4 implies that 2 maxf f ; gg f g belongs to L½a; b, and so maxf f ; gg belongs to R ½a; b. The second part of the assertion is proved similarly. Q.E.D. The Seminorm in L½a; b We will now define the ‘‘seminorm’’ of a function in L½a; b and the ‘‘distance between’’ two such functions. 10.2.9 Definition If f 2 L½a; b, we define the seminorm of f to be Z b k f k :¼ j f j: a If f ; g 2 L½a; b, we define the distance between f and g to be Z b distð f ; gÞ :¼ jj f gjj ¼ j f gj: a We now establish a few properties of the seminorm and distance functions. 10.2.10 Theorem The seminorm function satisfies: (i) (ii) (iii) (iv) jj f jj 0 for all f 2 L½a; b. If f ðxÞ ¼ 0 for x 2 ½a; b, then jj f jj ¼ 0. If f 2 L½a; b and c 2 R, then jjcf jj ¼ jcj jj f jj. If f ; g 2 L½a; b, then jj f þ gjj jj f jj þ jjgjj. Proof. Parts (i)–(iii) are easily seen. Part (iv) follows from the fact that j f þ gj j f jþ jgj and Theorem 10.1.5(c). Q.E.D. 10.2.11 Theorem The distance function satisfies: (i) (ii) (iii) (iv) distð f ; gÞ 0 for all f ; g 2 L½a; b. If f ðxÞ ¼ gðxÞ for x 2 ½a; b, then distð f ; gÞ ¼ 0. distð f ; gÞ ¼ distðg; f Þ for all f ; g 2 L½a; b. distð f ; hÞ distð f ; gÞ þ distðg; hÞ for all f ; g; h 2 L½a; b. These assertions follow from the corresponding ones in Theorem 10.2.10. Their proofs will be left as exercises. Using the seminorm (or the distance function) we can define what we mean for a sequence of functions ð f n Þ in L½a; b to converge to a function f 2 L½a; b; namely, given C10 12/09/2010 15:0:19 Page 307 10.2 IMPROPER AND LEBESGUE INTEGRALS 307 any e > 0 there exists KðeÞ such that if n KðeÞ then jj f n f jj ¼ distð f n ; f Þ < e: This notion of convergence can be used exactly as we have used the distance function in R for the convergence of sequences of real numbers. We will conclude this section with a statement of the Completeness Theorem for L½a; b (also called the Riesz-Fischer Theorem). It plays the same role in the space L½a; b that the Completeness Property plays in R. 10.2.12 Completeness Theorem A sequence ð f n Þ of functions in L½a; b converges to a function f 2 L½a; b if and only if it has the property that for every e > 0 there exists HðeÞ such that if m; n HðeÞ then jj f m f n jj ¼ distð f m ; f n Þ < e: The direction ()) is very easy to prove and is left as an exercise. A proof of the direction (() is more involved, but can be based on the following idea: Find a subsequence ðgk Þ :¼ ð f nk Þ of 1 X ð f n Þ such that jjgkþ1 gk jj < 1=2k and define f ðxÞ :¼ g1 ðxÞ þ ðgkþ1 ðxÞ gk ðxÞÞ, where k¼1 this series is absolutely convergent, and f ðxÞ :¼ 0 elsewhere. It can then be shown that f 2 L½a; b and that jj f n f jj ! 0. (The details are given in [MTI].) Exercises for Section 10.2 1. Show that Hake’s Theorem 10.2.1 can be given the following sequential formulation: A function f 2 R ½a; b if and only if there exists RA 2 R such that for any increasing sequence ðcn Þ in ða; bÞ c with cn ! b, then f 2 R ½a; cn and a n f ! A. 2. (a) Apply Hake’s Theorem to conclude that gðxÞ :¼ 1=x2=3 for x 2 ð0; 1 and gð0Þ :¼ 0 belongs to R ½0; 1. (b) Explain why Hake’s Theorem does not apply to f ðxÞ :¼ 1=x for x 2 ð0; 1 and f ð0Þ :¼ 0 (which does not belong to R ½0; 1). 3. Apply Hake’s Theorem to gðxÞ :¼ ð1 xÞ1=2 for x 2 ½0; 1Þ and gð1Þ :¼ 0. 4. Suppose that f 2 R ½a; c for all c 2 ða; bÞ and that there exists g 2 ða; bÞ and v 2 L½a; b such that j f ðxÞj vðxÞ for x 2 ½g; b. Show that f 2 R ½a; b. 5. Show that the function g1 ðxÞ :¼ x1=2 sinð1=xÞ for x 2 ð0; 1 and g1 ð0Þ :¼ 0 belongs to L½0; 1. (This function was also considered in Exercise 10.1.15.) 6. Show that the following functions (properly defined when necessary) are in L½0; 1. x ln x sin p x , , (b) (a) 1 þ x2 ln x ln x (c) ðln xÞðlnð1 xÞÞ, (d) pffiffiffiffiffiffiffiffiffiffiffiffiffi : 1 x2 7. Determine whether the following integrals are convergent or divergent. (Define the integrands to be 0 where they are not already defined.) Z 1 Z 1 sin x dx cos x dx , (b) , (a) x3=2 x3=2 0 0 Z 1 Z 1 ln x dx ln x dx pffiffiffiffiffiffiffiffiffiffiffiffiffi, (d) , (c) 2 0 x 1x 0 1x Z 1 Z 1 dx pffiffiffi (e) ðln xÞðsinð1=xÞÞ dx, (f) . xð1 xÞ 0 0 C10 12/09/2010 15:0:20 308 Page 308 CHAPTER 10 THE GENERALIZED RIEMANN INTEGRAL 8. If f 2 R½a; b, show that f 2 L½a; b. 9. If f 2 L½a; b, show that f 2 is not necessarily in L½a; b. 10. If f ; g 2 L½a; b and if g is bounded and monotone, show that f g 2 L½a; b. More exactly, if jg(x)j B, show that jj f gjj Bjj f jj. 11. (a) Give an example of a function f 2 R ½0; 1 such that max {f, 0} does not belong to R ½0; 1. (b) Can you give an example of f 2 L½0; 1 such that maxf f ; 0g 2 = L½0; 1? 12. Write out the details of the proof that minf f ; gg 2 R ½a; b in Theorem 10.2.8 when a f and a g. 13. Write out the details of the proofs of Theorem 10.2.11. 14. Give an f 2 L½a; b with f not identically 0, but such that jj f jj ¼ 0. 15. If f ; g 2 L½a; b, show that jj f jj jjgjj jj f gjj. 16. Establish the easy part of the Completeness Theorem 10.2.12. 17. If fn(x) :¼ xn for n 2 N, show that f n 2 L½0; 1 and that jj f n jj ! 0. Thus jjf n ujj ! 0, where u denotes the function identically equal to 0. 18. Let gn(x) :¼ 1 for x 2 [1, 1=n), let gn(x) :¼ nx for x 2 [1=n, 1=n] and let gn(x) :¼ 1 for x 2 (1/n, 1]. Show that jjgm gn jj ! 0 as m, n ! 1, so that the Completeness Theorem 10.2.12 implies that there exists g 2 L½1; 1 such that (gn) converges to g in L½1; 1. Find such a function g. 19. Let hn(x) :¼ n for x 2 (0, 1=n) and hn(x) :¼ 0 elsewhere in [0, 1]. Does there exist h 2 L½0; 1 such that jjhn hjj ! 0? 20. Let kn(x) :¼ n for x 2 (0, 1=n2) and kn(x) :¼ 0 elsewhere in [0, 1]. Does there exist k 2 L½0; 1 such that jjkn kjj ! 0? Section 10.3 Infinite Intervals In the preceding two sections, we have discussed the integration of functions defined on bounded closed intervals [a, b]. However, in applications we often want to integrate functions defined on unbounded closed intervals, such as ½a; 1Þ; ð1; b; or ð1; 1Þ: In calculus, the standard approach is to define an integral over [a, 1) as a limit: Z 1 Z g f :¼ lim f; a g!1 a and to define integrals over the other infinite intervals similarly. In this section, we will treat the generalized Riemann integrable (and Lebesgue integrable) functions defined on infinite intervals. In defining the generalized Riemann integral of a function f on [a, 1), we will _ :¼ adopt a somewhat different procedure from that in calculus. We note that if Q fð½x0 ; x1 ; t1 Þ; . . . ; ð½xn1 ; xn ; tn Þ; ð½xn ; 1; tnþ1 Þg is a tagged partition of [a, 1], then _ has the form: x0 ¼ a and xnþ1 ¼ 1 and the Riemann sum corresponding to Q ð1Þ f ðt1 Þðx1 x0 Þ þ þ f ðtn Þðxn xn1 Þ þ f ðtnþ1 Þð1 xn Þ: Since the final term f ðtnþ1 Þð1 xn Þ in (1) is not meaningful, we wish to suppress this term. We can do this in two different ways: (i) define the Riemann sum to contain only the first n terms, or (ii) have a procedure that will enable us to deal with the symbols 1 in calculations in such a way that we eliminate the final term in (1). C10 12/09/2010 15:0:21 Page 309 10.3 INFINITE INTERVALS 309 We choose to adopt method (i): instead of dealing with partitions of [a, 1) into a finite number of non-overlapping intervals (one of which must necessarily have infinite length), we deal with certain subpartitions of [a, 1), which are finite collections of non-overlapping intervals of finite length whose union is properly contained in [a, 1). We define a gauge on [a, 1] to be an ordered pair consisting of a strictly positive function d defined on [a, 1) and a number d > 0. When we say that a tagged subpartition P_ :¼ fð½x0 ; x1 ; t1 Þ; . . . ; ð½xn1 ; xn ; tn Þg is (d, d)-fine, we mean that ð2Þ ½a; 1Þ ¼ n [ ½xi1 ; xi [ ½xn ; 1Þ; i¼1 that ½xi1 ; xi ½ti dðti Þ; ti þ dðti Þ ð3Þ for i ¼ 1; . . . ; n; and that ½xn ; 1Þ ½1=d ; 1Þ ð4Þ or, equivalency, that 1=d xn : ð40 Þ Note Ordinarily we consider a gauge on [a, 1] to be a strictly positive function d with domain ½a; 1 :¼ ½a; 1Þ [ f1g where d(1) :¼ d. We will now define the generalized Riemann integral over [a, 1). 10.3.1 Definition (a) A function f : ½a; 1Þ ! R is said to be generalized Riemann integrable if there exists A 2 R such that for every e > 0 there exists a gaugede on [a, 1] _ A e. In this such that if P_ is any de-fine tagged subpartition of [a, 1), then Sð f ; PÞ case we write f 2 R ½a; 1Þ and Z 1 f :¼ A: a (b) A function f : ½a; 1Þ ! R is said to be Lebesgue integrable if both f and j f j belong to R ½a; 1Þ. In this case we write f 2 L½a; 1Þ. Of particular importance is the version of Hake’s Theorem for functions in R ½a; 1Þ. Other results for functions in L½a; 1Þ will be given in the exercises. 10.3.2 Hake’s Theorem If f : ½a; 1Þ ! R, then f 2 R ½a; 1Þ if and only if for every g 2 ða; 1Þ the restriction of f to [a, g] belongs to R ½a; g and Z ð5Þ lim g!1 Z In this case 1 a a g f ¼ A 2 R: f ¼ A. The idea of the proof of Hake’s Theorem is as before; the details are given in [MTI]. The generalized Riemann integral on the unbounded interval [a, 1) has the same properties as this integral on a bounded interval [a, b] that were demonstrated in Section 10.1. They can be obtained by either modifying the proofs given there, or by using Hake’s Theorem. We will give two examples. C10 12/09/2010 15:0:21 310 Page 310 CHAPTER 10 THE GENERALIZED RIEMANN INTEGRAL 10.3.3 Examples (a) If f ; g 2 R ½a; 1Þ, then f þ g 2 R ½a; 1Þ and Z 1 Z 1 Z 1 ð f þ gÞ ¼ fþ g: a a a If e > 0 is given let df be a gauge on [a, 1] such that if P_ is df-fine, then R Sð f ; PÞ _ 1 f e=2, and there exists a gauge dg such that if P_ is dg-fine, then Ra Sðg; PÞ _ 1 g e=2. Now let de ðtÞ :¼ minfdf ðtÞ; dg ðtÞg for t 2 ½a; 1 and argue as a in the proof of 10.1.5(b). (b) Let f : ½a; 1Þ ! R and let c 2 ða; 1Þ. Then f 2 R ½a; 1Þ if and only if its restrictions to [a, c] and [c, 1) are integrable. In this case, Z 1 Z c Z 1 ð6Þ f ¼ fþ f: a a c We will prove (() using Hake’s Theorem. By hypothesis, the restriction of f to [c, 1) is integrable. Therefore, Hake’s Theorem implies that for every g 2 ðc; 1Þ, the restriction of f to [c, g] is integrable and that Z 1 Z g f ¼ lim f: g!1 c c If we apply the Additivity Theorem 10.1.8 to the interval ½a; g ¼ ½a; c [ ½c; g, we conclude that the restriction of f to [a, g] is integrable and that Z g Z c Z g f ¼ fþ f; a whence it follows that Z g lim g!1 a Z f ¼ a c a c Z f þ lim g!1 c g Z f ¼ c a Another application of Hake’s Theorem establishes (6). Z fþ 1 c f: & 10.3.4 Examples (a) Let a > 1 and let fa(x) :¼ l=xa for x 2 ½1; 1Þ. We will show that f a 2 R ½1; 1Þ. Indeed, if y 2 ð1; 1Þ then the restriction of fa to [1, g] is continuous and therefore belongs to R ½1; g. Moreover, we have g Z g 1 1 1 1 1a x 1 dx ¼ ¼ : a 1a a1 g a1 1 x 1 But since the last term tends to l=(a 1) as g ! 1, Hake’s Theorem implies that f a 2 R ½1; 1Þ and that Z 1 1 1 when a > 1: dx ¼ a a1 1 x 1 X (b) Let ak be a series of real numbers that converges to A 2 R. We will construct a k¼1 function s 2 R ½0; 1Þ such that Z 1 0 s¼ 1 X k¼1 ak ¼ A: C10 12/09/2010 15:0:23 Page 311 10.3 INFINITE INTERVALS 311 Indeed, we define sðxÞ :¼ ak for x 2 ½k 1; kÞ, k 2 N. It is clear that the restriction of s to every subinterval [0, g] is a step function, and therefore belongs to R ½0; g . Moreover, if g 2 ½n; n þ 1Þ, then Z g s ¼ a1 þ þ an þ r g ; 0 where rg janþ1 j. But since the series is convergent, then rg ! 0 and so Hake’s Theorem 10.3.2 implies that Z g n X lim s ¼ lim ak ¼ A: g!1 0 n!1 k¼1 (c) If the function s is defined as in (b), then jsj has the value jak j on the interval 1 X ½k 1; kÞ; k 2 N. Thus s belongs to L½0; 1Þ if and only if the series jak j is convergent; 1 X k¼1 that is, if and only if ak is absolutely convergent. k¼1 (d) Let DðxÞ :¼ ðsin xÞ=x for x 2 ð0; 1Þ and let Dð0Þ :¼ 1. We will consider the important Dirichlet integral: Z 1 Z 1 sin x dx: DðxÞ dx ¼ x 0 0 Since the restriction R gof D to every interval ½0; g is continuous, this restriction belongs to R ½0; g . To see that 0 DðxÞdx has a limit as g ! 1, we let 0 < b < g. An integration by parts shows that Z g Z b Z g sin x dx DðxÞdx DðxÞdx ¼ x 0 0 b Z g cos x g cos x ¼ dx: x b x2 b But since jcos xj 1, it is an exercise to show that the above terms approach 0 as b < g tend to 1. Therefore the Cauchy Condition applies and Hake’s Theorem implies that D 2 R ½0; 1Þ. However, it will be seen in Exercise 13 that jDj does not belong to R ½0; 1Þ. Thus the & function D does not belong to L½0; 1Þ. We close this discussion of integrals over ½a; 1Þ with a version of the Fundamental Theorem (First Form). 10.3.5 Fundamental Theorem Suppose that E is a countable subset of ½a; 1Þ and that f, F : ½a; 1Þ ! R are such that: (a) F is continuous on ½a; 1Þ and lim F ðxÞ exists. x!1 = E. (b) F 0 ðxÞ ¼ f ðxÞ for all x 2 ða; 1Þ; x 2 Then f belongs to R ½a; 1Þ and Z 1 ð7Þ f ¼ lim F ðxÞ F ðaÞ: x!1 a Proof. If g is any number in ða; 1Þ, we can apply the Fundamental Theorem 10.1.9 to the interval [a, g] to conclude that f belongs to R ½a; g and Z g f ¼ F ðg Þ F ðaÞ: a C10 12/09/2010 15:0:23 312 Page 312 CHAPTER 10 THE GENERALIZED RIEMANN INTEGRAL Letting g ! 1, we conclude from Hake’s Theorem that f 2 R ½a; 1Þ and that equation (7) holds. Q.E.D. Integrals over ð1; b We now discuss integration over closed intervals that are unbounded below. Let b 2 R and g : ð1; b ! R be a function that is to be integrated over the infinite interval ð1; b. By a gauge on ½1; b we mean an ordered pair consisting of a number d > 0 and a strictly positive function d on ð1; bÞ. We say that a tagged subpartition P_ :¼ fð½x0 ; x1 ; t1 Þ; ð½x1 ; x2 ; t2 Þ; . . . ; ð½xn1 ; b; tn Þg of ð1; bÞ is ðd ; dÞ-fine in the case that n [ ð1; b ¼ ð1; x0 [ ½xi1 ; xi ; i¼1 that ½xi1 ; xi ½ti dðti Þ; ti þ dðti Þ for i ¼ 1; . . . ; n; and that ð1; x0 ð1; 1=d or, equivalently, that x0 1=d : Note Ordinarily we consider a gauge on ½1; b to be a strictly positive function d with domain ½1; b :¼ f1g [ ð1; b where dð1Þ :¼ d . n P Here the Riemann sum of g for P_ is S g; P_ ¼ gðti Þðxi xi1 Þ. i¼1 Finally, we say that g : ð1; b ! R is generalized Riemann integrable if there exists B 2 R such that for every e > 0 thereexists agauge de on ð1; b such that if P_ is _ any de -fine subpartition of ð1; b, then S g; P B e. In this case we write g 2 R ð1; b and Z b g ¼ B: 1 Similarly, a function g : ð1; b ! R is said to be Lebesgue integrable if both g and jgj belong to R ð1; b. In this case we will write g 2 Lð1; b. The theorems valid for the integral over ½a; 1 are obtained in this case as well. Their formulation will be left to the reader. Integrals over ð1; 1Þ Let h : ð1; 1Þ ! R be a function that we wish to integrate over the infinite interval ð1; 1Þ. By a gauge on ð1; 1Þ we mean a triple consisting of a strictly positive function d on ð1; 1Þ and two strictly positive numbers d ; d . We say that a tagged subpartition P_ :¼ fð½x0 ; x1 ; t1 Þ; ð½x1 ; x2 ; t2 Þ; . . . ; ð½xn1 ; xn ; tn Þg is ðd ; d; d Þ-fine in the case that ð1; 1Þ ¼ ð1; x0 [ n [ i¼1 ½xi1 ; xi [ ½xn ; 1Þ; C10 12/09/2010 15:0:24 Page 313 10.3 INFINITE INTERVALS 313 that ½xi1 ; xi ½ti dðti Þ; ti þ dðti Þ for i ¼ 1; . . . ; n; and that ð1; x0 ð1; 1=d and ½xn ; 1Þ ½1=d ; 1Þ or, equivalently, that x0 1=d and 1=d xn : Note Ordinarily we consider a gauge on [1, 1] to be a strictly positive function d with domain ½1; 1 :¼ f1g [ ð1; 1Þ [ f1g where dð1Þ :¼ d and dð1Þ :¼ d . n P Here the Riemann sum of h for P_ is S h; P_ ¼ hðti Þðxi xi1 Þ. i¼1 Finally, we say that h : ð1; 1Þ ! R is generalized Riemann integrable if there _ exists C 2 R such that for every e > 0 there exists a gaugede on [1, 1] such that if P is any de-fine subpartition of (1, 1), then S h; P_ C e. In this case we write h 2 R ð1; 1Þ and Z 1 h ¼ C: 1 Similarly, a function h : ð1; 1Þ ! R is said to be Lebesgue integrable if both h and jhj belong to R ð1; 1Þ. In this case we write h 2 Lð1; 1Þ. In view of its importance, we will state the version of Hake’s Theorem that is valid for the integral over (1, 1). 10.3.6 Hake’s Theorem If h : ð1; 1Þ ! R, then h 2 R ð1; 1Þ if and only if for every b < g in (1, 1), the restriction of h to [b, g] is in R ½b; g and Z g lim h ¼ C 2 R: Z In this case 1 1 b!1 g!þ1 b h ¼ C. As before, most of the theorems valid for the finite interval [a, b] remain true. They are proved as before, or by using Hake’s Theorem. We also state the first form of the Fundamental Theorem for this case. 10.3.7 Fundamental Theorem Suppose that E is a countable subset of (1, 1) and that h, H : ð1; 1Þ ! R satisfy: (a) H is continuous on (1, 1) and the limits lim HðxÞ exist. x! 1 = E. (b) H0 (x) ¼ h(x) for all x 2 ð1; 1Þ; x 2 Then h belongs to R ð1; 1Þ and Z 1 ð8Þ h ¼ lim H ðxÞ lim H ðyÞ: 1 x!1 y!1 C10 12/09/2010 15:0:26 314 Page 314 CHAPTER 10 THE GENERALIZED RIEMANN INTEGRAL 10.3.8 Examples (a) Let hðxÞ :¼ 1=ðx2 þ 1Þ for x 2 ð1; 1Þ. If we let HðxÞ :¼ Arctan x, then H 0 ðxÞ ¼ hðxÞ for all x 2 ð1; 1Þ. Further, we have lim HðxÞ ¼ x!1 1 1 2 p and lim HðxÞ ¼ 2 p. Therefore it follows that x!1 Z 1 1 x2 1 dx ¼ 12 p 12 p ¼ p: þ1 (b) Let kðxÞ :¼ jxjex for x 2 ð1; 1Þ. If we let KðxÞ :¼ 12 ð1 ex Þ for x 0 and 2 KðxÞ :¼ 12 ð1 ex Þ for x < 0, then it is seen that K is continuous on ð1; 1Þ and that 0 K ðxÞ ¼ kðxÞ for x 6¼ 0. Further, lim KðxÞ ¼ 12 and lim KðxÞ ¼ 12. Therefore it x!1 x!1 follows that 2 2 Z 1 1 2 jxjex dx ¼ 12 12 ¼ 1: & Exercises for Section 10.3 1. Let d be a gauge on [a, 1]. From Theorem 5.5.5, every bounded subinterval [a, b] has a d-fine partition. Now show that [a, 1] has a d-fine partition. 2. Let f 2 R ½a; g for all g a. Show that f 2 R ½a; 1Þ if and only if for every e > 0 there exists Rq KðeÞ a such that if q > p KðeÞ, then j p f j < e. 3. Let f and j f j belong to R ½a; g for all g a. Show that f 2R L½a; 1Þ if and only if for every q e > 0 there exists KðeÞ a such that if q > p > KðeÞ then p j f j < e. 4. Let f and R x j f j belong to R ½a; g for every g a. Show that f 2 L½a; 1Þ if and only if the set V :¼ a j f j : x a is bounded in R. R1 5. If f ; g 2 L½a; 1Þ, show that f þ g 2 L½a; 1Þ. Moreover, if jjhjj :¼ a jhj for any h 2 L½a; 1Þ, show that jj f þ gjj jj f jj þ jjgjj. 6. If f ðxÞ :¼ 1=x for x 2 ½1; 1Þ, show that f 2 = R ½1; 1Þ. 7. If f is continuous on ½1; 1Þ and if j f ðxÞj K=x2 for x 2 ½1; 1Þ, show that f 2 L½1; 1Þ. 8. Let f ðxÞ :¼ cos x for x 2 ½0; 1Þ. Show that f 2 = R ½0; 1Þ. 9. If s > 0, let gðxÞ :¼ esx for x 2 ½0; 1Þ. R1 (a) Use Hake’s Theorem to show that g 2 L½0; 1Þ and 0 esx dx ¼ 1=s. (b) Use the Fundamental Theorem 10.3.5. R1 10. (a) Use Integration by Parts and Hake’s Theorem to show that 0 xesx dx ¼ 1=s2 for s > 0. (b) Use the Fundamental Theorem 10.3.5. R1 11. Show that if n 2 N; s > 0, then 0 xn esx dx ¼ n!=snþ1 . R 1 1 12. (a) Show that the integral 1 R x ln x dx does not converge. 1 (b) Show that if a > 1, then 1 xa ln x dx ¼ 1=ða 1Þ2 . R ðnþ1Þp 1 jx sin xjdx > 1=4ðn þ 1Þ: 13. (a) Show that np (b) Show that jDj 2 = R ½0; 1Þ, where D is as in Example 10.3.4(d). R 1 pffiffiffi 14. Show that the integral 0 ð1= xÞsin x dx converges. [Hint: Integrate by Parts.] R1 15. Establish the convergence of Fresnel’s integral 0 sinðx2 Þdx. [Hint: Use the Substitution Theorem 10.1.12.] C10 12/09/2010 15:0:27 Page 315 10.4 CONVERGENCE THEOREMS 315 16. Establish the convergence or the divergence of the following integrals: Z 1 Z 1 ln x dx ln x dx pffiffiffiffiffiffiffiffiffiffiffiffiffi, , (b) (a) x2 þ 1 x2 þ 1 0 0 Z 1 Z 1 dx x dx , (d) , (c) xðx þ 1Þ ðx þ 1Þ3 0 0 Z 1 Z 1 dx Arctan x dx p ffiffiffiffiffiffiffiffiffiffiffiffiffi , (f) (e) . 3 3 x3=2 þ 1 1þx 0 0 17. Let f ; w : ½a; 1Þ ! R. Abel’s Test asserts that if f 2 R ½a; 1Þ and w is bounded and monotone on ½a; 1Þ, then f w 2 R ½a; 1Þ. R1 (a) Show that Abel’s Test does not apply to establish the convergencepofffiffiffi 0 ð1=xÞsin x dx by taking wðxÞ :¼ 1=x. However, it does apply if we take wðxÞ :¼ R1= x and use Exercise 14. 1 (b) Use Abel’s Test and Exercise 15 to show the convergence of R0 ðx=ðx þ 1ÞÞ sinðx2 Þ dx. 1 (c) Use Abel’s Test and Exercise 14 to show the convergence of 0 x3=2 ðx þ 1Þ sin x dx. (d) Use Abel’s Test to obtain the convergence of Exercise 16(f). 18. With the notation as inR Exercise 17, the Chartier-Dirichlet Test asserts that if f 2 R ½a; g for x all g a, if FðxÞ :¼ a f is bounded on ½a; 1Þ, and if w is monotone and lim wðxÞ ¼ 0, then x!1 f w 2 R ½a; 1. R1 (a) Show that Rthe integral 0 ð1=xÞ sin x dx converges. 1 (b) Show that 2 ð1=ln xÞ sin x dx converges. R 1 pffiffiffi (c) Show that 0 ð1= xÞ cos x dx converges. (d) Show Test does not apply to establish the convergence of R 1 that the Chartier-Dirichlet 2 2 0 ðx=ðx þ 1ÞÞ sinðx Þdx by taking f ðxÞ :¼ sinðx Þ. R 1 pffiffiffi 2 x sinðx Þdx is convergent, even though the integrand is not 19. Show that the integral 0 bounded as x ! 1. [Hint: Make a substitution.] 20. Establish the convergence of the following integrals: Z Z 1 ejxj dx, (b) (a) 1 Z 1 (c) 1 1 1 Z ex dx, 2 (d) ðx 2Þejxj dx, 1 2x dx . x ex e 1 Section 10.4 Convergence Theorems We will conclude our discussion of the generalized Riemann integral with an indication of the convergence theorems that are available for it. It will be seen that the results are much stronger than those presented in Section 8.2 for the (ordinary) Riemann integral. Finally, we will introduce a ‘‘measurable’’ function on ½a; b as the almost everywhere limit of a sequence of step functions. We will show that every integrable function is measurable, and that a measurable function on [a, b] is generalized Riemann integrable if and only if it satisfies a two-sided boundedness condition. We proved in Example 8.2.1(c) that if ð f k Þ is a sequence in R½a; b that converges on [a, b] to a function f 2 R½a; b, then it need not happen that Z b Z b f ¼ lim f k: ð1Þ a k!1 a However, in Theorem 8.2.4 we saw that uniform convergence of the sequence is sufficient to guarantee that this equality holds. In fact, we will now show that this is even true for a sequence of generalized Riemann integrable functions. C10 12/09/2010 15:0:28 316 Page 316 CHAPTER 10 THE GENERALIZED RIEMANN INTEGRAL 10.4.1 Uniform Convergence Theorem Let ð f k Þ be a sequence in R ½a; b and suppose that ð f k Þ converges uniformly on [a, b] to f. Then f 2 R ½a; b and (1) holds. Proof. Given e > 0, there exists KðeÞ such that if k KðeÞ and x 2 ½a; b, then we have j f k ðxÞ f ðxÞj < e. Consequently, if h; k KðeÞ, then 2e < f k ðxÞ f h ðxÞ < 2e Theorem 10.1.5 implies that Z 2eðb aÞ < a b Z fk for b a x 2 ½a; b: f h < 2eðb aÞ: Rb Since e > 0 is arbitrary, the sequence ð a f k Þ is a Cauchy sequence in R and therefore converges to some number, say A 2 R. We will now show that f 2 R ½a; b with integral A. For, if e > 0 is given, let KðeÞ be as above. If P_ :¼ fð½xi1 ; xi ; ti Þgni¼1 is any tagged partition of [a, b] and if k KðeÞ, then n X Sð f k ; PÞ Sð f ; PÞ ¼ f f k ðti Þ f ðti Þgðxi xi1 Þ i¼1 n X j f k ðti Þ f ðti Þjðxi xi1 Þ i¼1 n X eðxi xi1 Þ ¼ eðb aÞ: < i¼1 Rb Now fix r KðeÞ such that j a f r Aj < e and let dr;e be a gauge on [a, b] such that Rb _ < e whenever P_ is dr;e -fine. Then we have j a f r Sð f r ; PÞj Z b Z b Sð f ; PÞ A Sð f ; PÞ Sð f r ; PÞ þ Sð f r ; PÞ f r þ f r A a a < eðb aÞ þ e þ e ¼ eðb a þ 2Þ: Rb But since e > 0 is arbitrary, it follows that f 2 R ½a; b and a f ¼ A. Q.E.D. It will be seen in Example 10.4.6(a) that the conclusion of 10.4.1 is false for an infinite interval. Equi-integrability The hypothesis of uniform convergence in Theorem 10.4.1 is a very stringent one and restricts the utility of this result. Consequently, we now show that another type of uniformity condition can be used to obtain the desired limit. This notion is due to Jaroslav Kurzweil, as is Theorem 10.4.3. 10.4.2 Definition A sequence ð f k Þ in R ðIÞ is said to be equi-integrable if for every _ e > 0 there Rexists a gauge de on I such that if P is any de -fine partition of I and k 2 N, then Sð f k ; PÞ _ f < e. I k 10.4.3 Equi-integrability Theorem If ð f k Þ 2 R ðIÞ is equi-integrable on I and if f ðxÞ ¼ lim f k ðxÞ for all x 2 I, then f 2 R ðIÞ and Z Z ð2Þ f ¼ lim f k : I k!1 I C10 12/09/2010 15:0:29 Page 317 10.4 CONVERGENCE THEOREMS 317 Proof. We will treat the case I ¼ ½a; b; the general case can be found in [MTI]. Given e > 0, by the equi-integrability hypothesis, there exists a gauge de on I such that R _ f k < e for if P_ :¼ fð½xi1 ; xi ; ti Þgni¼1 is a de -fine partition of I, then we have Sð f k ; PÞ I all k 2 N. Since P_ has only a finite number of tags and since f k ðtÞ ! f ðtÞ for t 2 ½a; b, there exists a K e such that if h; k K e , then n X Sð f k ; PÞ Sð f h ; PÞ ð3Þ j f k ðti Þ f h ðti Þjðxi xi1 Þ eðb aÞ: i¼1 If we let h ! 1 in (3), we have Sð f k ; PÞ ð4Þ Sð f ; PÞ eðb aÞ for k Ke: Moreover, if h; k K e , then the equi-integrability hypothesis and (3) give Z Z Z f k f h f k Sð f k ; PÞ þ Sð f k ; PÞ Sð f h ; PÞ I I I Z þSð f h ; PÞ f h e þ eðb aÞ þ e ¼ eð2 þ b aÞ: I R Since e > 0 is arbitrary, then ð I f k Þ is a Cauchy sequence and converges to some A 2 R. If we let h ! 1 in this last inequality, we obtain Z f k A eð2 þ b aÞ for k K e : ð5Þ I We now show that f 2 R ðIÞ with integral A. Indeed, given e > 0, if P_ is a de -fine partition of I and k K e , then Z Z Sð f ; PÞ A Sð f ; PÞ Sð f k ; PÞ þ Sð f k ; PÞ f k þ f k A I eðb aÞ þ e þ eð2 þ b aÞ ¼ eð3 þ 2b 2aÞ; I where we used (4) for the first term, the equi-integrability for the second, and (5) for the third. Since e > 0 is arbitrary, f 2 R ðIÞ with integral A. Q.E.D. The Monotone and Dominated Convergence Theorems Although the Equi-integrability Theorem is interesting, it is difficult to apply because it is not easy to construct the gauges de . We now state two very important theorems summarizing the most important convergence theorems for the integral that are often useful. McLeod [pp. 96–101] has shown that both of these theorems can be proved by using the Equiintegrability Theorem. However, those proofs require a delicate construction of the gauge functions. Direct proofs of these results are given in [MTI], but these proofs also use results not given here; therefore we will omit the proofs of these results. We say that a sequence of functions on an interval I R is monotone increasing if it satisfies f 1 ðxÞ f 2 ðxÞ f k ðxÞ f kþ1 ðxÞ for all k 2 N; x 2 I. It is said to be monotone decreasing if it satisfies the opposite string of inequalities, and to be monotone if it is either monotone increasing or decreasing. 10.4.4 Monotone Convergence Theorem Let ð f k Þ be a monotone sequence of func tions in R ðIÞ such that f ðxÞ ¼ lim R f k ðxÞ almost everywhere on I. Then f 2 R ðIÞ if and only if the sequence of integrals I f k is bounded in R, in which case Z Z ð6Þ f ¼ lim f k : I k!1 I C10 12/09/2010 15:0:30 318 Page 318 CHAPTER 10 THE GENERALIZED RIEMANN INTEGRAL The next result is the most important theorem concerning the convergence of integrable functions. It is an extension of the celebrated ‘‘Lebesgue Dominated Convergence Theorem’’ from which it can also be proved. 10.4.5 Dominated Convergence Theorem Let ð f n Þ be a sequence in R ðIÞ and let f ðxÞ ¼ lim f k ðxÞ almost everywhere on I. If there exist functions a; v in R ðIÞ such that ð7Þ aðxÞ f k ðxÞ vðxÞ for almost every x 2 I; then f 2 R ðIÞ and Z ð8Þ I Z f ¼ lim k!1 I f k: Moreover, if a and v belong to LðIÞ, then f k and f belong to LðIÞ and Z ð9Þ jj f k f jj ¼ j f k f j ! 0: I Note If a and v belong to LðIÞ, and we put w :¼ maxfjaj; jvjg, then w 2 LðIÞ and we can replace the condition (7) by the condition ð70 Þ j f k ðxÞj wðxÞ for almost every x 2 I: Some Examples 10.4.6 Examples (a) If k 2 N, let f k ðxÞ :¼ 1=k for x 2 ½0; k and f k ðxÞ :¼ 0 elsewhere in ½0; 1Þ. R 1 Then the sequence converges uniformly on ð0; 1Þ to the 0-function. However, 0 f k ¼ 1 for all k 2 N, while the integral of the 0-function equals 0. It is an exercise to show that the function supf f k ðxÞ : k 2 Ng does not belong to R ½0; 1Þ, so the domination condition (7) is not satisfied. Z 1 k x þ1 dx ¼ 13 : (b) We have lim k!1 0 xk þ 3 For, if gk ðxÞ :¼ ðxk þ 1Þ=ðxk þ 3Þ, then 0 gk ðxÞ 1 and gk ðxÞ ! 1=3 for x 2 ½0; 1Þ. Thus the Dominated Convergence Theorem 10.4.5 applies. Z k xk ax 1 if a > 1. 1þ e dx ¼ (c) We have lim k!1 0 k a1 Let hk ðxÞ :¼ ð1 þ x=kÞk eax for x 2 ½0; k and hk ðxÞ :¼ 0 elsewhere on ð0; 1Þ. The argument in Example 3.3.6 shows that ðhk Þ is an increasing sequence and converges to ex eax ¼ eð1aÞx on ½0; 1Þ. If a > 1 this limit function belongs to L½0; 1Þ. Moreover, if FðxÞ :¼ eð1aÞx =ð1 aÞ, then F 0 ðxÞ ¼ eð1aÞx so that the Monotone Convergence Theorem 10.4.4 and the Fundamental Theorem 10.3.5 imply that 1 Z 1 Z 1 1 ð1aÞx : hk ¼ e dx ¼ FðxÞ ¼ lim k!1 a1 0 0 0 (d) If f Ris bounded and continuous on ½0; 1Þ and if a > 0, then the function defined by 1 LðtÞ :¼ 0 etx f ðxÞdx is continuous for t 2 J a :¼ ða; 1Þ. Since jetx f ðxÞj Meax for t 2 J a, if ðtk Þ is any sequence in J a converging to t0 2 J a , the Dominated Convergence Theorem implies that Lðtk Þ ! Lðt0 Þ. But since the sequence ðtk Þ ! t0 is arbitrary, then L is continuous at t0 . C10 12/09/2010 15:0:32 Page 319 10.4 CONVERGENCE THEOREMS 319 (e) The integral in (d) is differentiable for t > a and Z 1 0 ð10Þ L ðtÞ ¼ ðxÞetx f ðxÞdx; 0 which is the result obtained by ‘‘differentiating under the integral sign’’ with respect to t. Fix a number t0 2 J a. If t 2 J a , then by the Mean Value Theorem applied to the function t 7! etx , there exists a point tx between t0 and t such that we have etx et0 x ¼ xetx x ðt t0 Þ, whence tx e et0 x xetx x xeax : tt 0 Since vðxÞ :¼ xeax f ðxÞ belongs to L½0; 1Þ, then for any sequence ðtk Þ in J a with t0 6¼ tk ! t0 , the Dominated Convergence Theorem implies that Z 1 Lðtk Þ Lðt0 Þ etk x et0 x lim lim ¼ f ðxÞ dx k!1 tk t0 tk t0 0 k!1 Z 1 ðxÞet0 x f ðxÞ dx: ¼ 0 0 Since ðtk Þ is an arbitrary sequence, then L ðt0 Þ exists and (10) is proved. Z k tx sin x e (f) Let Dk ðtÞ :¼ dx for k 2 N; t 0: x 0 Since jðetx sin xÞ=xj etx 1 for t 0; x 0, the integral defining Dk exists. In particular, we have Z k sin x dx: Dk ð0Þ ¼ x 0 We Dk ð0Þ ! 12 p as k ! 1. By Example 10.3.4(d), this will show that R 1 want to show that 1 0 ðsin xÞ=x dx ¼ 2 p. The argument is rather complex, and uses the Dominated Convergence Theorem several times. tx @ e sin x ¼ jetx sin xj 1 for t 0, Since the partial derivative satisfies @t x x 0, an argument as in (e) and the Dominated Convergence Theorem imply that Z k D0 k ðtÞ ¼ etx sin x dx for k 2 N; t 0: 0 @ etx ðt sin x þ cos xÞ Since a routine calculation shows that ¼ etx sin x, then an @x t2 þ 1 application of the Fundamental Theorem gives D0 k ðtÞ ¼ etk ðt sin k þ cos kÞ 1 2 : t2 þ 1 t þ1 etk ðt sin k þ cos kÞ for 0 t k and gk (t) :¼ 0 for t > k, then another t2 þ 1 application of the Fundamental Theorem gives Z k Z k Z k dt ð11Þ D0 k ðtÞdt ¼ gk ðtÞdt Dk ðkÞ Dk ð0Þ ¼ 2þ1 t 0 0 0 Z 1 gk ðtÞdt Arctan k: ¼ If we put gk ðtÞ :¼ 0 C10 12/09/2010 15:0:32 320 Page 320 CHAPTER 10 THE GENERALIZED RIEMANN INTEGRAL If we note that gk(t) ! 0 for t > 0 as k ! 1 and that (since k 1) etk ðt þ 1Þ 2e1 for t 0; t2 þ 1 R1 then the Dominated Convergence Theorem gives 0 gk ðtÞdt ! 0. In addition, since j(sin x)=xj 1, we have Z k Z k x¼k ekx kx sin x kx dx e e dx ¼ jDk ðkÞj ¼ x k x¼0 0 0 2 1 ek 1 ! 0: ¼ k k jgk ðtÞj Therefore, as k ! 1, formula (11) becomes 0 lim Dk ð0Þ ¼ 0 lim Arctan k ¼ 12 p: k!1 k!1 As we have noted before, this gives an evaluation of Dirichlet’s Integral: Z 1 sin x 1 dx ¼ p: ð12Þ x 2 0 & Measurable Functions We wish to characterize the collection of functions in R ðIÞ. In order to bypass a few minor details, we will limit our discussion to the case I :¼ [a, b]. We need to introduce the notion of a ‘‘measurable function’’; this class of functions contains all the functions the reader is ever likely to encounter. Measurable functions are often defined in terms of the notion of a ‘‘measurable set.’’ However, the approach we will use is somewhat simpler and does not require a theory of measurable sets to have been developed first. (In fact, the theory of measure can be derived from properties of the integral; see Exercises 15 and 16.) We recall from Definition 5.4.9 that a function s : ½a; b ! R is a step function if it has only a finite number of values, each value being assumed on a finite number of subintervals of [a, b]. 10.4.7 Definition A function f : ½a; b ! R is said to be (Lebesgue) measurable if there exists a sequence (sk) of step functions on [a, b] such that ð13Þ f ðxÞ ¼ lim sk ðxÞ k!1 for almost every x 2 ½a; b: We denote the collection of all measurable functions on [a, b] by M½a; b. Z ð14Þ We can reformulate the definition as: A function f is in M½a; b if there exists a null set [a, b] and a sequence (sk) of step functions such that f ðxÞ ¼ lim sk ðxÞ k!1 for all x 2 ½a; bnZ: It is trivial that every step function on [a, b] is a measurable function. By Theorem 5.4.10, a continuous function on [a, b] is a uniform limit of a sequence of step functions; therefore, every continuous function on an interval [a, b] is measurable. Similarly, every monotone function on [a, b] is a uniform limit of step functions (see the proof of Theorem 7.2.8); therefore, every monotone function on an interval is measurable. C10 12/09/2010 15:0:33 Page 321 10.4 CONVERGENCE THEOREMS 321 At first glance, it might seem that the collection of measurable functions might not be so very large. However, the requirement that the limit (13) is required to hold only almost everywhere (and not everywhere), enables one to obtain much more general functions. We now give a few examples. 10.4.8 Examples (a) The Dirichlet function, f (x) :¼ 1 for x 2 [0, 1] rational and f (x) :¼ 0 for x 2 [0, 1] irrational, is a measurable function. Since Q \ ½0; 1 is a null set, we can take each sk to be the 0-function. We then obtain sk(x) ! f (x) for x 2 ½0; 1nQ. (b) Thomae’s function h (see Examples 5.1.6(h) and 7.1.7) is a measurable function. Again, take sk to be the 0-function. Then sk(x) ! h(x) for x 2 ½0; 1nQ. (c) The function g(x) :¼ 1=x for x 2 (0, 1] and g(0) :¼ 0 is a measurable function. This can be seen by taking a step function sk(x) :¼ 0 for x 2 [0,1=k) and (using 5.4.10) such that jsk(x) 1=xj < 1=k for x 2 [1=k, 1]. Then sk(x) ! g(x) for all x 2 [0, 1]. (d) If f 2 M½a; b and if c : ½a; b ! R is such that cðxÞ ¼ f ðxÞ a.e., then c 2 M½a; b. For, if f (x) ¼ lim sk(x) for x 2 [a, b]nZ1 and if c(x) ¼ f (x) for all x 2 [a, b]nZ2, then c(x) ¼ lim sk(x) for all x 2 [a, b]n(Z1 [ Z2). Since Z1 [ Z2 is a null set when Z1 and Z2 are, the conclusion follows. & The next result shows that elementary combinations of measurable functions lead to measurable functions. 10.4.9 Theorem Let f and g belong to M½a; b and let c 2 R. (a) Then the functions cf, j f j, f þ g, f g, and f g also belong to M½a; b. (b) If w : R ! R is continuous, then the composition w f 2 M½a; b. (c) If ( fn) is a sequence in M½a; b and f (x) ¼ lim fn (x) almost everywhere on I, then f 2 M½a; b. Proof. (a) We will prove that j f j is measurable. Let Z [a, b] be a null set such that (14) holds. Since jskj is a step function, the Triangle Inequality implies that 0 jj f ðxÞj jsk ðxÞjj j f ðxÞ sk ðxÞj ! 0 for all x 2 [a, b]nZ. Therefore j f j 2 M½a; b. The other assertions in (a) follow from the basic properties of limits. (b) If sk is a step function on [a, b], it is easily seen that w sk is also a step function on [a, b]. Since w is continuous on R and f (x) ¼ lim sk(x) for all x 2 [a, b]nZ, it follows that ðw f ÞðxÞ ¼ wð f ðxÞÞ ¼ lim wðsk ðxÞÞ ¼ limðw sk ÞðxÞ for all x 2 [a, b]nZ. Therefore w f is measurable. (c) This conclusion is not obvious; a proof is outlined in Exercise 14. Q.E.D. The next result is that we can replace the step functions in Definition 10.4.7 by continuous functions. Since we will use only one part of this result, we content ourselves with a sketch of the proof of the other part. 10.4.10 Theorem A function f : ½a; b ! R is in M½a; b if and only if there exists a sequence (gk) of continuous functions such that ð15Þ f ðxÞ ¼ lim gðxÞ k!1 f or almost every x 2 ½a; b: C10 12/09/2010 15:0:33 322 Page 322 CHAPTER 10 THE GENERALIZED RIEMANN INTEGRAL Proof. (() Let Z [a, b] be a null set and (gk) be a sequence of continuous functions such that f (x) ¼ lim gk(x) for x 2 [a, b]nZ. Since gk is continuous, by 5.4.10 there exists a step function sk such that jgk ðxÞ sk ðxÞj 1=k for all x 2 ½a; b: Therefore we have 0 j f ðxÞ sk ðxÞj j f ðxÞ gk ðxÞj þ j gk ðxÞ sk ðxÞj j f ðxÞ gk ðxÞj þ 1=k; whence it follows that f (x) ¼ lim gk(x) for all x 2 [a, b]nZ. Sketch of ()) Let Z be a null set and (sk) be a sequence of step functions such that f (x) ¼ lim sk(x) for all x 2 [a, b]nZ. Without loss of generality, we may assume that each sk is continuous at the endpoints a, b. Since sk is discontinuous at only a finite number of points in (a, b), which can be enclosed in a finite union Jk of intervals with total length 1=k, we can construct a piecewise linear and continuous function gk that coincides with sk on [a, b]nJk. It can be shown that gk(x) ! f (x) a.e. on I. (See [MTI] for the details.) Q.E.D. Functions in R ½a; b are Measurable We now show that a generalized Riemann integrable function is measurable. 10.4.11 Measurability Theorem If f 2 R ½a; b, then f 2 M½a; b. Proof. Let F : ½a; b þ 1 ! R be the indefinite integral Z FðxÞ :¼ x a f if x 2 ½a; b; and let F(x) :¼ F(b) for x 2 (b, b þ 1]. It follows from the Fundamental Theorem (Second Form) 10.1.11(a) that F is continuous on [a, b]. From 10.1.11(c), there exists a null set Z such that the derivative F 0 (x) ¼ f (x) exists for x 2 [a, b]nZ. Therefore, if we introduce the difference quotient functions gk ðxÞ :¼ Fðx þ 1=kÞ FðxÞ 1=k for x 2 ½a; bÞ; k 2 N; then gk(x) ! f (x) for all x 2 [a, b]nZ. Since the gk are continuous, it follows from the part of Theorem 10.4.10 we have proved that f 2 M½a; b. Q.E.D. Are Measurable Functions Integrable? Not every measurable function is generalized Riemann integrable. For example, the function g(x) :¼ 1=x for x 2 (0,1] and g(0) :¼ 0 was seen in Example 10.4.8(c) to be measurable; however, it is not in R ½a; b because it is ‘‘too large’’ (as x ! 0 þ). However, if the graph of a measurable function on [a, b] lies between two functions in R ½a; b, then it also belongs to R ½a; b. C10 12/09/2010 15:0:34 Page 323 10.4 CONVERGENCE THEOREMS 323 10.4.12 Integrability Theorem Let f 2 M½a; b. Then f 2 R ½a; b if and only if there exist functions a; v 2 R ½a; b such that ð16Þ aðxÞ f ðxÞ vðxÞ f or almost every x 2 ½a; b: Moreover, if either a or v belongs to L½a; b, then f 2 L½a; b: Proof. ()) This implication is trivial, since one can take a ¼ v ¼ f . (() Since f 2 M½a; b, there exists a sequence (sk) of step functions on [a, b] such that (13) holds. We define sk :¼ midfa; sk ; vg for k 2 N, so that sk ðxÞ is the middle of the numbers a(x), sk(x), and v(x) for each x 2 [a, b]. It follows from Theorem 10.2.8 and the facts midfa; b; cg ¼ minfmaxfa; bg; maxfb; cg; maxfc; agg; minfa0 ; b0 ; c0 g ¼ minfminfa0 ; b0 g; c0 g; that sk 2 R ½a; b and that a sk v. Since f ¼ lim sk ¼ lim sk a.e., the Dominated Convergence Theorem now implies that f 2 R ½a; b. If either a or v belongs to L½a; b, then we can apply Theorem 10.2.6 to conclude that f belongs to L½a; b. Q.E.D. A Final Word In this chapter we have made frequent reference to Lebesgue integrable functions on an interval I, which we have introduced as functions in R ðIÞ whose absolute value also belongs to R ðIÞ. While there is no single ‘‘standard approach’’ to the Lebesgue integral, our approach is very different from any that are customary. A critic might say that our approach is not useful because our definition of a function in LðIÞ is not standard, but that would be wrong. After all, one seldom uses the definition to confirm that a specific function is Lebesgue integrable. Instead, one uses the fact that certain simpler functions (such as step functions, polynomials, continuous functions, bounded measurable functions) belong to LðIÞ, and that more complicated functions belong to LðIÞ by taking algebraic combinations or various limiting operations (e.g., Hake’s Theorem or the Dominated Convergence Theorem). A famous analyst once said, ‘‘No one ever calculates a Lebesgue integral; instead, one calculates Riemann integrals and takes limits.’’ It is the same as with real