MATH 3210 Syllabus, Fall 2013
September 1, 2013
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General Information
• Course: Foundations of Analysis I (Math 3210, Section 001).
• Description: We will study single-variable calculus using rigorous, proof-based mathematical arguments. Key topics are the real numbers, limits, continuous functions,
differentiation, integration, and series. Unlike previous calculus courses, the emphasis
is not on computation, but on learning how to understand and explain concepts in a
logical and complete manner. These skills are essential in all higher-level, proof-based
mathematics courses.
• Prerequisites: Officially,“C” or better in ((MATH 2210 OR MATH 1260 OR MATH
1280 OR MATH 1321) AND (MATH 2200 OR MATH 2270 OR MATH 2250)). Practically, you should be familiar with basic logic (at the level of http://www.math.utah.edu/
%7Earoberts/M3210-1d.pdf), as well as sets, functions, and mathematical proof (see
sections 1.1, 1.2, and 1.3 of Taylor).
• Instructor: Thomas Goller. You can call me “Thomas”.
• Course Webpage: Google my name and follow the links. I will post all updates on
the webpage, so check it frequently!
• Time and Place: M,W,F 8:05 AM - 9:25 AM; JWB 208.
• Textbook:
Chapters 1-6 of Foundations of Analysis,
ISBN: 9780821889848. I’m expecting you to read every word.
by Joseph Taylor,
• Homework: Weekly problem sets consisting mostly of exercises from the book. Please
work in groups and ask me questions, but you absolutely must write up your own
solutions! Start working on the exercises as early in the week as possible.
• Midterms: Ugly, repulsive things!
• Learning Celebrations (LCs): There will be five learning celebrations, at the beginning of class on the Monday of every third week. See the timeline below.
• Final Learning Celebration (FLC): Monday, December 16, 2013 from 8-10 AM in
the usual room.
• Grading: 30% Homework, 40% LCs, 30% FLC. I am hoping the department will
provide a grader to grade your homework and provide feedback. I will grade the LCs
and the FLC. I’ll assign letter grades using a curve, so forget about that awful “≥ 95%
is an ‘A’, don’t you dare make any mistakes!” nonsense. I will report average scores for
homework and LCs as we proceed and give you a rough idea of how scores correspond
to letter grades. The average grade will likely be around a B-, though it could be
higher if average scores are high.
• Office Hours: Right after class, or e-mail me to set up an appointment. You can also
just drop by my office, JWB 307, at the risk that I’ll be busy right at that moment.
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Rough Course Timeline
Dates
Reading
Topics
8/26, 8/28, 8/30
1.4, 1.5
Intro, real numbers
LD, 9/4, 9/6
2.1
Limits
HW 1 due 9/4
9/9, 9/11 , 9/13
2.2, 2.3
Limit theorems
HW 2 due 9/9
9/16, 9/18, 9/20
2.4, 2.5, 2.6
Monotone, Cauchy, limsup
HW 3 due 9/16
9/23, 9/25, 9/27
3.1, 3.2
Continuity
HW 4 due 9/23
9/30, 10/2 , 10/4
3.3, 3.4
Uniform
HW 5 due 9/30
10/7, 10/9, 10/11
4.1, 4.2
Derivative
HW 6 due 10/7
10/21, 10/23, 10/25
4.3, 4.4
MVT, L’Hopital
HW 7 due 10/21
10/28, 10/30 , 11/1
5.1, 5.2
Integral
HW 8 due 10/28
11/4, 11/6, 11/8
5.3, 5.4
FTC, more on integration
HW 9 due 11/4
11/11, 11/13, 11/15
6.1, 6.2
Series convergence, tests
HW 10 due 11/11
11/18, 11/20 , 11/22
6.3, 6.4
Power series
HW 11 due 11/18
11/25, 11/27, TG
6.5
Taylor’s formula
HW 12 due 11/25
12/2, 12/4, 12/6
Leftover material
HW 13 due 12/2
12/9, 12/11 , 12/13
Review
HW 14 due 12/9
12/16, 8-10 AM
Final LC
Assignments
FB
The boxed dates are the dates of the five learning celebrations.
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3
Reading the Book
I’ve had a lot of practice reading proof-based math textbooks, and I regret to inform you
that it’s really hard. Math textbooks are often a rapid-fire collection of definitions, theorems,
and proofs, with barely any pictures, motivation, and examples. It is not unusual for me
to spend an hour poring over a single page of a textbook. Here is some advice for tackling
definitions and theorems.
Definitions
Some definitions are simply new terminology for an old idea; these shouldn’t give you any
trouble. Definitions that present new concepts can be incredibly difficult to digest (try 2.1.4
at your own risk!). Unfortunately, you will be using these difficult definitions in proofs,
so you will have to master them. A possible procedure for mastering a definition is the
following:
(Step 1) Scan the definition. If the definition uses a previously defined term whose meaning
you have forgotten, review that term.
(Step 2) Read the definition carefully and try to capture the “intuition” behind the definition. Use words, not symbols. The word chosen for the new term can be helpful.
(Step 3) Write down some examples that satisfy the definition, and some non-examples.
Start with examples that are as simple as possible and gradually make them more
intricate! These examples can help you with Step 2.
(Step 4) Know how to prove and disprove all conditions and properties that appear in the
definition.
(Step 5) Think about the definition in a broader context. Why is this term defined the way
it is? Can it be generalized?
If you’re lucky, the book you are reading will take care of some of these steps for you. If the
book gives examples, come up with a few of your own as well. If you can’t come up with
examples to illustrate a concept, then you are far from having even a basic understanding
of that concept!
Example. Let’s apply the above procedure to a simple definition: An integer n is even if
there is an integer k such that n = 2k.
(Step 1) The only previously-defined concept you have to know is “integer”.
(Step 2) The definition is saying that the integer n is a multiple of 2.
(Step 3) Some easy examples of even integers are 2 = 2 · 1, 4 = 2 · 2, and 6 = 2 · 3. Can
negative integers be even? Sure: −2 = 2 · (−1), −4 = 2 · (−2), and so on. Is 0
even? Yes: 0 = 2 · 0. Some integers that are not even are 1, −1, 3, and −3. Every
number that is not an integer cannot be even since the definition applies only to
integers.
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(Step 4) To prove an integer n is even, you have to find an integer k such that n = 2k. For
instance, 6 is even because 6 = 2 · 3. To prove an integer is not even, you have to
show that n #= 2k for all integers k (once you know that every non-even integer is
odd, you can just prove that n is odd).
(Step 5) Think about why “even” is defined only for integers. You could extend the definition
to rational numbers by saying “A rational number x is even if there is a rational
number y such that x = 2y.” Similarly, you could define “even” for real numbers.
Do these definitions give you the integer definition of “even” when you view the
integers as a subset of the rational or real numbers? Why are these extensions not
very useful?
Example. Now let’s try a more challenging definition (Taylor 1.5.2): Let A be a non-empty
subset of R. We define the supremum of A, denoted sup A, to be the smallest extended real
number M such that a ≤ M for every a ∈ A.
(Step 1) You may have to review “extended real number”. The extended real numbers are
just the usual real numbers together with ∞ and −∞. ∞ is ≥ every real number,
while −∞ is ≤ every real number.
(Step 2) The condition “a ≤ M for every a ∈ A” means that M is greater than or equal to
every element of A, namely M is an upper bound for A or M equals ∞. If A has
a maximum element, then that element is the supremum.
(Step 3) If A = [0, 1], then sup A = 1 since 1 is the maximum element of A. If A = (0, 1),
then A has no maximum element, but we still have sup A = 1. If A = (0, 1) ∪ 2,
then sup A = 2. If A = (−3, ∞), then A has no upper bounds, so sup A = ∞. I
can’t think of any interesting non-examples.
(Step 4) If A has no upper bounds, then sup A = ∞. If A does have upper bounds, then to
prove that an extended real number M equals sup A, you have to show two things:
• M is an upper bound for A, namely a ≤ M for every a ∈ A.
• M is the smallest upper bound for A. This can be achieved by showing either
that (1) if N is any upper bound for A, then M ≤ N , or that (2) any number
smaller than N , which we can write as N − ! for some real number ! > 0, is
not an upper bound for A, namely there is an a ∈ A such that a > N − !.
To prove that a real number M is not equal to sup A, you can show either that M
is not an upper bound for A, or that there is a number smaller than M that is an
upper bound for A.
(Step 5) Some things to think about: Why do we need A to be non-empty? If we apply
the definition to the empty set, we seem to get sup ∅ = −∞. Why don’t we want
this? (Hint: Theorem 1.5.7 (a).) Why do we need the notion of supremum when
we are also defining “maximum” (p. 28)? Does the concept of supremum work in
the integers or rational numbers?
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Theorems
You will need to invoke theorems in the book in the proofs you write. You must master anything you use in a proof because “small” mistakes in a proof can be devastating. Fortunately,
theorems are a bit easier to master than difficult definitions. Try the following:
(Step 1) Scan the theorem. If the theorem uses a term whose meaning you have forgotten,
review that term.
(Step 2) Read the theorem carefully. If the theorem is a mess of symbols, try to state the
main idea in words.
(Step 3) Check the theorem on some examples to make sure you’re understanding it correctly. If the theorem has a lot of conditions, try to come up with non-examples that
relax as few conditions as possible. These non-examples will help you understand
why all the conditions in the theorem are necessary.
Steps 1-3 are all you really need to successfully apply the theorem in your proofs. If you
want to really understand why the theorem is true, you’ll have to grapple with its proof.
Some tips for doing this are:
(Step 4) Try to prove the theorem on your own. You will probably fail, but you may be
able to identify some of the difficulties in proving the theorem, which will help you
understand the proof in the book.
(Step 5) Skim the proof of the theorem and try to extract the general idea. Return to Step
4.
(Step 6) Keep repeating Steps 4 and 5 until you prove the theorem. If you get stuck, work
through the proof line-by-line, making sure you understand every step, and noting
where each assumption in the theorem is being used.
(Step 7) Come back to the theorem at a later time and repeat the process starting with Step
4. The proof should come more easily each time you repeat this cycle.
By repeating the proof of a difficult theorem, you will better absorb the key ideas in the
proof. This is desirable because key ideas in proofs tend to recur in other proofs.
Example. Consider Theorem 1.5.4 (a): Let A be a non-empty subset of R and let x be an
element of R. Then sup A ≤ x if and only if a ≤ x for every a ∈ A.
(Step 1) Make sure you know what “sup A” means.
(Step 2) The theorem is saying that for a real number x, the condition of being at least as
big as sup A is equivalent to being an upper bound for A.
(Step 3) If A = R, then sup A = ∞, so the set of real numbers ≥ sup A is empty. Likewise,
there are no upper bounds for A. If A = (0, 1), then sup A = 1. Thus the set of real
numbers that are ≥ sup A is [1, ∞), which is the same as the set of upper bounds
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for A. A similar analysis holds when A = (0, 1]. One condition we could try to
relax is A being non-empty, but sup ∅ is not defined, so the theorem wouldn’t even
make any sense. If we were to define sup ∅ = −∞, then the theorem would hold
for A = ∅.
(Step 4) If you’ve mastered the definitions, you should be able to prove this theorem. Here’s
the idea. If A is unbounded, then the theorem is trivial because there are no real
numbers ≥ ∞ and no upper bounds for A. When A is bounded, the proof boils
down to the fact that sup A is the smallest upper bound for A. No tricks required,
but you should write out the proof carefully, splitting the “if and only if” into two
parts and carefully unraveling the definitions. The book gives a thorough proof.
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What to Expect from Lecture
As an undergraduate math major, I sat through many math lectures that looked roughly
like this:
• For the first five minutes, the instructor reviews the previous lecture. I studied that
material after the previous lecture, and this review solidifies my understanding.
• In the next ten minutes, the instructor introduces a chain of new definitions, with a
few examples mixed in. The definitions and examples seem to make sense.
• Now the instructor starts proving some theorems involving the new definitions. Since
a proof requires board space, those definitions quickly vanish from the board. Good
thing I copied them down in my notes.
• For the next ten minutes my attention frantically jumps between the definitions in my
notes and the proof the instructor is presenting. I miss most of the logic of the proof,
and I still haven’t absorbed the new definitions.
• For the last half hour of lecture, I listen to the instructor speak, hoping to absorb some
knowledge without having a clue what he or she is saying.
Lectures like this are not the ideal vehicle for teaching mathematics. Unfortunately, such
lectures naturally emerge when instructors have a lot of material to cover and aren’t getting
enough feedback from students.
Here are some measures I’m planning to avoid such dreary lectures.
(1) Write key definitions on side boards to preserve them.
(2) Do a thorough analysis of difficult definitions and theorems (using the steps above).
(3) Allocate 15-30 minutes of each lecture for working on problems in pairs.
You can help:
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(1) Begin to master the definitions in the new section before lecture (use the above 5-step
process). You should expect to get stuck – that’s what lecture is for. The important
thing is that you are not seeing the definitions for the first time in lecture.
(2) Please ask questions and answer my questions during lecture. Questions and answers
help me pace the lecture, and the more engaged you are, the more you will learn and
the more fun lecture will be for me.
(3) Be willing to discuss new material and work on problems with a partner.
By working in pairs, you also have the opportunity to get to know everyone in the class who
regularly attends lectures. I recommend working with a different partner each lecture. Try
sitting at a different desk each lecture, so that someone in an adjacent desk is more likely to
become a new partner.
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The Instructor’s Role
For mathematics courses at the 3000 level and above, you will need to be strongly selfmotivated. The material is too rich and difficult for an instructor to teach everything in
lecture. As a result, you will have to read the book carefully and spend hours working
on exercises. My role as the instructor is to help you and guide you in your study of the
material. My lectures will not be comprehensive, but I’ll still expect you to read all the
material in the book and work on exercises that may be unlike anything you have seen in
lecture. I will present the material I consider most important or especially difficult.
Ultimately, you are responsible for ensuring this course is useful to you. You will need
to work hard, be awake in lecture, and get help when you need it. Falling behind is not an
option because the material in this course is extremely cumulative!
I am happy to meet with you when you have questions. I’m in my office most of the day.
Please come prepared for a discussion, not a lecture. I’d be delighted to give you hints on
homework problems, but I won’t work out every detail of every homework problem.
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Homework
Homework is being assigned for you. Some of the key benefits of homework are to:
• Give you an incentive to grapple with the material and collaborate with other students.
• Give you practice writing mathematics.
• Give you feedback on your writing.
Please write your solutions neatly. Your proofs should be neat, clear, and concise. Study the
proofs in the book as models. Think of a proof as a short essay, which should be succinct and
complete but well-written. Your target audience is a mathematics student who knows all
the definitions and theorems we’ve covered, and is familiar with standard methods of proof
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(direct, contrapositive, contradiction, induction), but doesn’t know how to prove the claim
you’re proving. Write proofs clearly using complete English sentences that may contain some
symbols.
Example. Here is an example of a claim (admittedly, a simple one) and a neat, clear, concise
proof.
Theorem. Let n ∈ Z. If n is even, then n3 is even.
Proof. Since n is even, there is k ∈ Z such that n = 2k. Then n3 = (2k)3 = 2(4k 3 ), so n3 is
even.
One of the most important skills for writing proofs is knowing how to use definitions.
You shouldn’t state definitions in your proof (I didn’t write “An integer n is even if there is
an integer k such that n = 2k”), but you should very clearly indicate where definitions are
being used in the logical argument (“Since n is even”). You also have to know how to prove
that a mathematical object satisfies a given definition. In the example, to show n3 is even,
we need to find some integer j such that n3 = 2j. In this case, j = 4k 3 made the proof work.
Let me know if the homework assignments are too long, short, easy, or difficult. I want
homework to be as useful as possible for you, and you can help make it so.
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About Mathematics
Mathematics is a game. The initial position of the game is a list of axioms that do not
contradict each other, plus a list of logical rules for constructing proofs. In the solitaire
version of the game, you have three possible moves:
(1) Make a new definition, possibly referring to existing definitions.
(2) Make a new statement about existing definitions.
(3) Prove or disprove an existing statement. A statement that has been proven is called a
“theorem”, while a statement that has been disproven is discarded or modified until it
can be proven.
You keep making moves with the goal of amassing as many interesting definitions and theorems as possible!
Here is an example of how the game could begin. My first axiom is the existence of the
set of integers with addition and multiplication operations that satisfy the usual properties
(identity, associativity, commutativity, distributivity) and the usual order relation (<, ≤, >,
≥). My second axiom is the following theorem (which can actually be proved from the first
axiom):
Theorem. Let n be an integer and let d be an integer such that d > 0. Then there are
unique integers q, r with 0 ≤ r < d such that
n = dq + r.
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I’ll use the usual logical rules. Here are some possible moves:
Definition 1. An integer n is blue if there is an integer k such that n = 3k. An integer
n is green if there is an integer k such that n = 3k + 1. An integer n is red if there is an
integer k such that n = 3k + 2. In each case, we say that “blue”, “green”, or “red” is the
color of the integer.
Theorem 2. The integers 0, 3, and 6 are blue. The integers 1, 4, and 7 are green. The
integers 2, 5, and 8 are red.
Proof. We can write
0 = 3 · 0,
3 = 3 · 1,
6 = 3 · 2,
1 = 3 · 0 + 1,
4 = 3 · 1 + 1,
7 = 3 · 2 + 1,
2 = 3 · 0 + 2,
5 = 3 · 1 + 2,
8 = 3 · 2 + 2.
Theorem 3. Every integer has exactly one color.
Proof. Let n be an integer. By the second axiom, there are are unique integers q, r with
0 ≤ r < 3 such that
n = 3q + r.
Since r must be 0, 1, or 2, we see that n has a color, and since r is unique, n cannot have
more than one color.
Definition 4. An integer n is called thick if there is an integer " such that n = "2 . An
integer that is not thick is called thin.
Theorem 5. 0, 1, and 4 are thick. 2 and 3 are thin.
Proof. Since 0 = 02 , 1 = 12 , and 4 = 22 , all three of these integers are thick. Now we will
prove that 2 and 3 are thin by showing that k 2 cannot be equal to 2 or 3 for any integer
k. First, if k ≥ 2 or k ≤ 2, then k 2 ≥ 4, so k 2 #= 2 and k 2 #= 3. Now we check the three
remaining possibilities for k. If k = 0, then k 2 = 0. If k = 1 or k = −1, then k 2 = 1. Thus
k 2 #= 2 and k 2 #= 3 for every integer k.
Theorem 6. Let n be a thick integer. Then n is blue or green.
Proof. Since n is thick, there is an integer " such that n = "2 . Now " has a color, so we just
need to check that n is blue or green in three cases:
• If " is blue, then " = 3k for some integer k. Then n = "2 = (3k)2 = 3(3k 2 ), so n is
blue.
• If " is green, then " = 3k + 1 for some integer k. Then n = "2 = 9k 2 + 6k + 1 =
3(3k 2 + 2k) + 1, so n is green.
• If " is red, then " = 3k + 2 for some integer k¿ Then n = "2 = 9k 2 + 12k + 4 =
3(3k 2 + 4k + 1) + 1, so n is green.
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Wow, I managed to define the color of an integer and thick and thin integers, and prove
that thick integers cannot be red!
The solitaire version of mathematics is fun because you have so much freedom. Unfortunately, it is easy to run out of ideas, and other people will have to retrace your entire
game if they want to understand your results! For these reasons, the collaborative version
of mathematics is far more popular. Although you could play collaborative mathematics
with a small group of friends, why not join the global game that has been running for many
years?
To join the global game of mathematics, one must use the same list of axioms and the
same logical rules that everyone else is using. One can make the same moves as in the
solitaire version, plus a few others:
(4) Read a paper or book published by another player to get ideas. You can use his or her
definitions and theorems in your own work.
(5) Publish your own work in a mathematics journal. This allows other players to use your
work.
Moves of type (4) are crucial because the global game has already been running for quite
some time, so you’ll have some catching up to do. Moves of type (5), which many players
are eager to make, are difficult because the only way your work will get published is if it is
interesting to other players. To be interesting, it’ll have to be original and substantial.
Let’s play! In Foundations of Analysis I, we’ll begin our entry into the global game of
mathematics by making an important move of type (4). We’ll carefully read the first six
chapters of Taylor’s book to get a firm grasp of the definitions and theorems he presents.
These definitions and theorems are powerful and other players refer to them frequently. We’ll
also practice making moves of type (3) in the context of our new definitions and theorems.
This course will be helpful if you’re planning to be a casual player who focuses on moves of
type (4), and absolutely essential if you are dreaming of making a move of type (5) someday.
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