xiv
Contents
365
Mathematical Induction
C.I
C.2
C.3
C.4
C.5
Three Equivalent Statements
The Principle of Mathematical Induction
The Principle of Strong Induction . . . .
The Well-Ordering Property
Some Comments on Induction Arguments
1
365
367
371
374
376
Real Numbers
Bibliography
377
Index
379
Where should this book begin? Every author faces this question ;is
before an empty piece of paper or a blank computer screen. h>i
important to catch the reader's attention and create a desire lo conlmn.
a mathematics textbook, it is generally assumed that the reader ; i h r . n l \
motivation to study the given subject—perhaps an intrinsic inleivsi 01
requirement. The more relevant question is what background in .1
mathematical knowledge does the reader possess as he or she o|vir, i l n , I
the first time? A novel that is boring w i l l collect dust on a s h e l l . .1 m . i i h . i
textbook that is confusing will experience a similar laic.
The assumed background for a reader of t i n s textbook is ;i » m n | > l i <
sequence and some degree of mathematical sophistication. W i n k t i n I
is rather ambiguous, it essentially means that the reader undeisl.md'. i l >
proofs in mathematics, is willing to attempt in read and i i i i d r r . l . i m l | i n »
is able to think abstractly. In addition, the u-adri need', i < > \\.\\<•
knowledge of the following topics:
1. sels and operations on sets;
2. l i n n lions and properties of l i n u l i o n .
V null
I
LJcni
( mil.
ill' il in.In. lion.
'• |
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li il.
i-
M< h i
• i.i
jiionl li\ i nllll.iclli I
iK'i'p
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ii
Mil
V
ll
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.1 '.lllll'l'lr
In > l l l i l l | i > | ,
{ . , , | |,i mill'
Mi h . n I III i l l i . in il I
. .1 III llll
I,.|IHI,
I ii ml
illiill
HI.I
ill i i | < l i n
i i . i l . i n . i l \ . 1 . i ' . l l i c ' . l i n l s nl n . i l i i i i i n l i i i
• I
nl i i . i l n i i m l ' i i
.mil
11 i i i i i i m i . de l i n e d m i SCtN < > l i f . 11 i i m i i l " , r. 11 i . | > n > l i . i l ' l \ .1 " i m i l u l r . i In I n " i n \\ i l l i
H i ' i l r l l l l l l l i i l i nl .1 n . i l i i i i m h r i
Tins may
. . T I I I l i k e - M i l nnncccssai V | > l . n r I n ' . I . i l l
a i n o n e l e a d i n j ' l l n s honk hiis been workinj; w i l l i real numbers lui M Ic.r.i
A l l l i o n j ' h Ilk 1 q u e s t i o n , "What is a real number?", may appeal In
N i * an ( i b v i t u i s answer, an allempl lo give a precise answer to t h i s question o l i e n
I. ids lo circular definitions or an appeal lo common sense. (For an interestin<.> and
i . i ii' discussion concerning the fouiidation for a course in real analysis, sec
h u m i.in | . ' ( ) | . ) As w i t h many objects in this remarkable world, close scrutiny leads
i n . n 11, i/1 n ; ' a n d a j ' i ' i aval ing pu/./.les. Atoms, cells, and galaxies are much richer t h a n
a n s u n e ever could have imagined, and the same is true of the set of real numbers.
In l l n s chapter, we w i l l examine the concept of a real number and set the stage for
the lest dl the material in t h i s book.
I.I
length ol' llic hypotenuse is not a rational number
\ \ l n r l i are used for counting objects. The natural numbers can also be associated
w i t h the passage of time; one day follows another just as 2 follows 1, 3 follows
' .ind so on. Some people will argue that these numbers are hard-wired into the
h u m a n brain while others claim the natural numbers are a property of the universe
in w i n c h we live, but we will not enter the realms of psychology or philosophy here.
Hie intrui'i-s are the numbers
HI M i n . in everyday life, a great deal of elementary school mathematics is devoted
n i i l y of these numbers.
Since there are an i n f i n i t e number of rational numbers and (more importantly)
..... there is another rational number between any two distinct rational numbers,
II would appear that the rational numbers are suitable for all possible measure...... is. This "optimistic" view of the rational numbers is easily dispelled. By the
r , ihagorean Theorem, the hypotenuse of a right triangle with sides of length 1 has
.1 I. i r ' i h of -v/2 (see Figure 1.1). The ancient Greek mathematicians knew that this
l n " i h could not be described by dividing a stick into equal parts and taking some
number of the parts. In other words, the number \/2 is not rational. Although some
n .ulers may be familiar with a proof that \/2 is not rational, a proof is given in the
n e x t paragraph.
The proof is an example of a proof by contradiction. Suppose that \/2 is a
rational number. Then \/2 = p/q, where p and q are positive integers that have no
common divisors greater than 1. It follows that p2 = 2q2, which indicates that p2
is an even integer. The only way for p2 to be even is for p to be even. Let p = 2r,
where r is a positive integer, and compute
. . . , -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5 , . . . ,
2q2 = p2 = (2r) 2 = 4r 2 .
\\ l u i h extend indefinitely in either direction. In this context, the natural numbers are
ued lo as the positive integers. The negative integers are introduced to solve
pioblems, such as x + 5 = 2, for which there are no solutions in the set of positive
Inti "ITS. We will use the symbol Z to represent the set of integers and the symbol
1
lo denote the set of positive integers. The symbol Z comes from the German
\ \ n u l lor number, Zdhlen.
This implies that q1 = 2r2, so q is also an even integer. But p and q cannot both be
even integers since they have no common divisors greater than 1 . This contradiction
proves that \/2 is not a rational number.
There are many other "lengths" that cannot be represented as the ratio of two
integers. It follows that the set of rational numbers is not big enough for the purpose
of exact measurement. Consequently, it is necessary to introduce a larger set of
numbers known as the set of real numbers and denoted by the symbol R. Every
rational number is a real number, and the real numbers that are not rational numbers
are called irrational numbers. In a nutshell, the real numbers contain all of the
numbers needed in the development of calculus. The rational numbers have a simple
description as the ratio of two integers but, as we will see, there is not an equally
simple description of the set of real numbers.
Two of the familiar ways to describe a real number are the following.
I.I
WHAT Is A REAL NUMBER?
i in luliiral
numbers (or the counting numbers) arise, well, quite naturally. These
Ifi tin 1 numbers
1,2,3,4,5,6,7,8,9, 10,...,
When numbers are used for measurement, it is sometimes necessary to consider
i MI ul a whole. This need for parts or fractions of an integer leads to the concept of
n i in. i n , i l number. A rational number is a number of the form /»/</. where p and q
me iiiic)'.crs with q ^ 0. In other words, a rational number is a ratio of two integers.
i i i . i.mdaid symbol for the set of rational numbers is the symbol <(.]), where the
'I the Idler Q follows from the fact that rational numbers are quotients. The
M i i i i i l i e i t/5 can be interpreted as follows: divide a slick i n t o *i equal pieces and
hike I ol the pieces. The rational numbers include the i n l e c e i s since an integers
'" i l .o lie represented as n/\. Since the rational number. ,ne ihe l a m i l i a r numbers
1. A real number is a decimal expansion.
2. A real number represents a point on a number line.
4
ii
1 ( 1 ( 1 , rvrniiiiilly one
l">il,o|imir
\^
"""II)' \\l\\\ III,.',,
1
'
I. < i i M . i l , s|>.m .mil i
I ' III
I,
M ill Ml < a l.-\\
,,| ,1, , in, , 1 ,
i
|,n
.„ 1 1 1 , , | | N ,m inlnnir N
I l,
|||
| !„,,(,|,
I MI HI.I.UM ,
iih, i ,|in, i i \ In thin particular euse) and start a repeat ing
ilmi)' h.i|i|n ii 1 . \\ lirn ,ni\ i. ihoM.il mmibri is converted to a decimal.
' ih, iiiiink i i, |>n ., ni, , I \t\ ( I n - decimal
m h,
ih,
M
O.I Ml Ml M I 4 3 I 4 3 . . . .
O.I.MI-MI MI M
I illl llii
Ml
III,
mmili, i \ .111(1 compute
.HIM
I
II)
_
__
__
104
105
106
Mm '" r l r m c i i i a r y knowledge of i n f i n i t e series is required U> duly understand
i , i l n i i i i i l i c i s expressed in iliis w;iy. A second problem is dial (he integer 10 plays
I kc) i n k - m decimal expansions, hut il is certainly possible to represent a real
i i i i m b r t using powers of integers other than 10; common examples are 2, 16, and 60.
l l u w c v r i , a real iiinnher is independent of the base (the integer whose powers are
il . , 1 ) in which il is represented. Even with decimal expansions, the representation
ul ,i n-al number need not be unique. For example,
0.2000000... = 0.1999999....
I- ice i his, letjc = 0.1999999... and compute
o.v = 10* -x = 1.9999999... -0.1999999... = 1.8.
11 follows that x = 0.2. These difficulties do not indicate that decimal expansions
IK invalid or useless, but they do indicate that decimal expansions are not suitable
lor a mathematical definition of a real number.
In terms of decimal expansions, there is a simple distinction between rational
numbers and irrational numbers: a real number is a rational number if and only if
ils decimal expansion has a repeating pattern. Rather than give a general proof of
t h i s result, we will illustrate it with two examples. Consider the rational number
15/101. To convert this number to a decimal, we must perform long division:
liii)i>t
,
|,n. 1 4 . ^ 1 4 3 1 4 3 . . . -0.143143143... = 143,
Illi d m , l i , , i i i - s i i i . i i v ; 143/999. A similar computation is possible for any
M i l e x p a n s i o n t h a i develops a repeating pattern. The number
O.I 0100 1 000 10000 100000 10000001 ...
i 111 i n a l i u n a l number since its decimal expansion has no repeating pattern (the
...... il>er ul O's between 1's continues to increase). Note that the key here is the
i ir/n'ii/ini;. The preceding decimal expansion has a pattern, but it is not a
ilinjj, pattern. The fact that decimal expansions of irrational numbers have no
i , I" .ilinjj, pattern reveals another problem. In terms of decimal expansions, what
i i l n - number \/2? Since this number is irrational, its decimal expansion has no
i , i" a i i u g pattern. Consequently, there are real numbers whose complete decimal
1
I lansions are unknown.
The second interpretation of a real number is geometrical in nature. In this
we mark off two points on a straight line and call the left point 0 and the right
I I U M I I I . All other real numbers then represent points on the line. The number 1/2
is the point midway between 0 and 1 , the number 2 is twice as far to the right of 0
as 1 , and the number \/2 represents the length of the hypotenuse of a right triangle
w i t h sides of length 1 . Negative numbers appear to the left of 0 using the same unit
ul measurement. A portion of the number line appears below.
1/2
.148514-
101
15.000000-••
10 1
4 90
4 04
860
808
520
505
150
101
490
11 follows that
15
ToT
= 0.1485148514851 I K .
()ne of the obvious problems with this interpretation of a real number is that operations such as multiplication and division are difficult to perform. In addition, the
point on the line that represents a real number varies with the choice of 0 and 1, but
the concept of a real number should not depend on a given spatial configuration.
In other words, real numbers should be independent of geometry. Finally, there is
a philosophical problem with the number line. A point has no length, a line has
length, and a line is made up of points. How can you combine "things" that do not
have a property and end up with something that suddenly has that property? You
may recall that some of Zeno's paradoxes (see Katz [11] for a discussion of these)
are concerned with this problem. (Zeno of Elea was a Greek philosopher who lived
in the fifth century B.C.E.)
The last few paragraphs are not intended to convince the reader to stop thinking
of a real number as a decimal expansion or as a point on a line. These are very useful
interpretations of a real number, but they are not suitable for a rigorous definition
(l
I1
I I \MiMl I .. Ki ill N
di .1 ir.il muni., i \ \ i i n i i i .
clcllnUion of anew ic
uiiirm.n
in I h r . CAie, I l i e n i l
| ,1, i i m i h . i i ! > ! i mil milliner'/ i i "
> ,.u, |,>, M..H ly dollneil li
mil,
,ll l l l i i n l i r i ' , .in- (lie
'kiii>\Hi
i|iianlilu".
A i c . i l m i n i U ' i is
i l r l l l i r c l Id IK- Sdini 1 surl ol dhjei I i n v o l v m ; 1 llic i . i l i o i u l i i i i i n l i c i s eilhci ,i scl nl
l a l i d i i a l i i n m l u T s ( k n o w n as Dodckind i-uls) or ;is equivalence classes i > l ( 'anchy
•.ei|neiKvs ol' rational luimlKTs (sec Ilir next cliaplci loi Ihe i l r l m i i i o n ol a ( ' a n c h y
sequence). Tin" operations of addition and n u i l l i p l i c - a l i o n arc then defined on these
objects and all ol'lhc usual properties of the real mimluTS follow as theorems. This
piocess is nol all thai diflicnll, hul it is (odious and roi|uiivs somo mathematical
so|)liislioalion. Depending on your personality, il is oilhor a fascinaling procoss or a
poinlloss oxoroiso. In any case, wo are nol going lo include this development here.
The reader can consul! R u d i n |22| for a dcvelopmenl of the real numbers using
I )edckind cuts; a thorough account of both definitions can be found in Hobson 11()|.
In between the useful bul problematic familiar interpretations of a real number
and (he rigorous bul very abstract interpretations of a real number is an interpretation
lhal we will use for (his book. Consider the following formula, which illustrates the
distributive property of the real numbers:
n(x + a) = nx + na.
While working with a formula of this type, it is unlikely that the symbols n, x,
and a are considered to be decimal expansions, points on a line, or some type of
sel composed of rational numbers. The symbols are just objects—namely, real
numbers—that satisfy the given property. In other words, in most manipulations
involving real numbers, the focus is on the properties of the real numbers rather than
on some particular interpretation of the real numbers. Il is possible to give a short
list of properties of the real numbers from which all of the others follow. A real
number is then considered to be a member of a set of objects lhal has all of these
properties. The purpose of the next few paragraphs is to list those properties.
If the reader is encountering abstract mathematics for ihe lirsi time, then the
next paragraphs will probably seem rather mysterious. If you are in this position,
then as you read through the following material. Ihink about Ihe usual properties
of numbers that you have used over the years and nole how each numbered item
in the various definitions lists one of those piopeiiu-s. Malhcmaticians seek a firm
logical foundation for the terms and symbols t l u - \ use, the concept of an ordered
Held is such a foundation for the sel ol i r a l numbers In order to work through
Ihis text, it is important that the readci he I. l i . n w i t h Ihe properties of the set of
real numbers. However, a compleic nndri .i.iiiilmj' ol ihe formal definition of the
sel of real numbers is not crucial lot Ilir i c m a i i n l c i of ihe text. (By the way, the
ideas presented here will make
sum level of mathematical maturity
increases.)
DEFINITION 1.1 A field i
dolinedonit. These opeiain >n
in the usual way. Addii
1. x + y € I loi . i l l \
.1 m
n
I i.
i . /
2. jc + y = v I i IIH ill .
ipi\ M-I /•' of objects that has two operations
> . | i In ion and multiplication and are denoted
. m m salisl'y the following properties:
y i /
»
M
|
l I
I
I
I
M
I
I Idl .ill \ . V, Z 6
INT.'
F.
i /
..I
ii i i MII ii ihai v (- 0 = x for all x e F.
i . l i t . / • Mine r\isis v i /• such that x + y — 0.
i. • \ . / loi all * , v e F.
7. ( Y
H. i \\ ').•
v \ lor all \ , v (_ /•'.
\ ( v . : ) for all*, y, z e F.
'i. I contains an element 1 such that x • 1 = x for all x e F.
10. I''01 each v i /•' such that x ^ 0 there exists y € F such that xy = 1.
1 1 . \ ( v I ;:) = xy + xz for all x, y, z e F.
The held properties are very familiar properties of the real numbers and are
UNi'd correctly by most people most of the time with very little conscious thought.
I 'i i >| iort ies ( I ) and (6) assert that the set F is closed under addition and multiplication,
lhal is, Ihe sum and product of two elements in the field are elements of the field.
1'iopeiiies (2), (3), (7), and (8) are the commutative and associative properties
dl addition and multiplication, respectively. Properties (4) and (9) establish the
. M'.ience of additive and multiplicative identities, while properties (5) and (10)
establish the existence of additive and multiplicative inverses. Finally, property (11)
is Ihe distributive property.
Il is worth noting that our definition of a field makes a number of assumptions.
;
I or instance, it is assumed that the reader knows what an operation is, is familiar
with the notation for addition and multiplication, and understands the order in which
operations are to be performed. To list and define all of these assumptions would
lake us too far afield, so we will leave such a discussion to another book. The
interested reader can consult the abstract algebra texts by Birkhoff and MacLane [2]
or Gallian [7]. The set of rational numbers is one example of a field, but there are
other examples as well; most texts on abstract algebra will contain many examples
of fields.
DEFINITION 1.2 An order < on a set 5 is a relation that satisfies the following
two properties:
1. If x, y e S, then exactly one of x < y, x — y, or y < x is true.
2. For all x, y, z e S, if x < y and y < z, then x < z.
An ordered set is a set with an order defined on it.
Although the term "relation" has a precise mathematical definition, it will not
be given here. It is sufficient to think of a relation as a relationship between two
numbers. For this informal presentation, we are assuming that the reader is familiar
with comparing the sizes of real numbers. For real numbers x and y, the inequality
x < v means that the number x is less than the number y. As an example, the set
of integers with the symbol m < n meaning (as usual) that m is less than n is an
ordered set.
i
m-MIMM ION I t
\n nnli ii <Mi
>M /
lllil
(I, then \ I \-
')
ill t i n
lolllIU I I I ! ' . l l l l l l l M l l l . i l |l|il|l ( I I I .
I. If v • 0 and v
liinlHli \nliM, I ill i1 null, mill li)i'i|iiiillllii
I W. I i'l ( In mi liiiilliiiiiil iiuMilii i I'mvi lliiil llieie e x i s t s .111 irrnlicMi.il iiiiinhei v such that
(I
2. If x > 0 and y > 0, then xy > 0.
3. x < y if and only if y — x > 0.
For the record, there are other ways to define an ordered field, and even w i i h i n
the approach that has been adopted here, there are variations. In fact, Definition 1.3
involves a degree of redundancy: it is possible to prove that property (3) implies
property (1). However, the three properties listed in Definition 1.3 are very familialproperties: the sum of two positive numbers is positive, the product of two positive
numbers is positive, and x is less than y if and only if y — x is positive.
All of the familiar properties of equalities and inequalities are valid in an ordered
field. As a typical example, the property,
if x < y and z > 0, then xz < yz,
follows from the properties of an ordered field. We will neither state nor prove
these results here and simply assume that the reader is familiar with these properties
of real numbers. In any case, proving these properties is not the purpose of this
textbook. Once again, the interested reader is referred to a book on abstract algebra.
The two most familiar examples of ordered fields are the set of rational numbers
and the set of real numbers. (As an aside, the set of complex numbers is a field that
is not an ordered field.) In other words, the field properties and the order properties
do not distinguish between the sets Q and ffi; both sets satisfy all of the properties
that have been listed thus far. Since the sets Q and K have some differences, the
real numbers must possess some additional property that the rational numbers do
not possess. A discussion of the distinction between these two sets of numbers will
be the topic of Section 1 .3.
\ V I'l II Illlllllllll Illimhrl
III, I el \ I n - i i ii'nl iiiiinl)i-i
iinilii'ii.il
I ' n i v e ih.ii .ii least niie of the numbers \/2 — x or \/2 + x is
I I . I el n be a positive integer thai is not a perfect square. Prove that Jn is irrational.
}. I'love i h a t \/2 + -</3 is irrational.
,1, 1'iove I hat s//i — I + \/n + 1 is irrational for every positive integer n.
I'tove I hat there is no rational number r such that 2r = 3.
Ifjt I el > and y he irrational numbers such that x — y is also irrational. Define sets A and
/( by A - |.\ f A- : r 6 Q} and B = {y + r : r e Q } . Prove that the sets A and B
have no elements in common.
PA. I ,iM A he the set of all numbers of the form a + b^/2, where a and b are arbitrary
lational numbers. Let addition and multiplication be defined on A in the same way
they are delined for real numbers. Prove that the set A is a field.
||7. I el I! be the set of all irrational numbers together with the numbers 0, 1, and — 1. Let
addition and multiplication be defined on B in the same way they are defined for real
numbers. Determine the field properties that are satisfied by B. Is B a field?
|H. I ,el S be the set of all ordered pairs of positive integers and adopt the convention that
(11, /;) = (c, d) if and only if a = c and b = d. For each given relation, determine
whether or not the relation satisfies the properties listed in Definition 1.2.
a) Define a relation < on S by (a, b) < (c, d) if and only if ab < cd.
b) Define a relation < on S by (a, b) < (c, d) if and only if ad < be.
I'). Referring to Definition 1.3, prove that property (3) implies property (1). Be certain to
use only properties that have been proved or already assumed to be true.
10. Use the properties of an ordered field to prove the following: if x < y and z > 0, then
xz < yz.
Exercises
1.2
lj It was stated in the text that the rational numbers form a field. In particular, the rational
numbers are closed under addition and multiplication. Prove this fact by showing that
the sum of two rational numbers is a rational number and the product of two rational
numbers is a rational number.
Prove that there is a rational number between any two distinct rational numbers.
Convert each of the rational numbers into a repeating decimal.
a) 8/27
b) 4/21
c) 5/19
p4J Convert each repeating decimal into a rational number of the form />/</, where p and
' q are positive integers with no common divisors.
a) 0.357357357...
b) 0.327272727...
c) 0.21 15.W.IM5.W46. ..
('s. Find the millionth digit in the decimal expansion of 2/7.
A. Prove that the reciprocal of an irrational number is an irrational I I I H I I | > C T .
7. Prove that the sum of a rational number and an irrational nnml>ri is m.iiioiwl.
X. irrational
Prove thatnumber.
the product of a nonzero rational number and an u i . i i i o i i a l number is an
ABSOLUTE VALUE, INTERVALS, AND INEQUALITIES
This section contains a discussion of the absolute value function, the concept of an
interval, a formula for geometric sums, and a brief exploration of two interesting
inequalities. The properties of the absolute value function will be used throughout
the text, primarily as a measure of the distance between two numbers. Almost
all the theorems discussed in this book are stated for functions that are defined
on an interval, so it is important to understand exactly what is meant by the term
"interval". The formula for a geometric sum is quite simple and appears now and
again in proofs and examples. Although the last two inequalities discussed in this
section will not be used in this text, they are important in other areas of real analysis
and are related to ideas that have been considered thus far.
In analysis, it is often necessary to measure the distance between two points.
Since the points considered in analysis may represent numbers, vectors, functions,
or sets, the notion of distance between points and how to compute it can become
rather abstract. It is possible to study this notion in a more general setting (see the
discussion of metric spaces in Section 8.5), but at this stage of the game, we will
remain in the familiar territory of the real numbers. The absolute value function can