All Course Documents

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Thanks to all for a great semester!

Course Description

Broadly speaking, real analysis is a rigorous, extended version of calculus. This semester, you will study numbers, functions, continuity, and differentation in a new light. We will answer such questions as:

• Where does mathematics "begin"?
• Are some infinite sets "larger" than others?
• Can a sequence of numbers be rearranged to converge to a different number?
• How many discontinuities can a function have?
• You may understand 1, 2, and 3 dimensional objects. But what about 1.5-dimensional objects?
• What do voting, donuts, the streets in Manhattan, and Facebook have in common?

Answering these questions requires the rigorous, axiomatic approach to definition, theorem, and proof, which is the foundation of theoretical mathematics. Along the way, you will pick up more proof-writing skills, and also skills that enable you to better read, understand, and communicate mathematics. I will also describe how the field of analysis fits into the broader "picture" of mathematics.

Textbook

The textbook for the course is Real Mathematical Analysis by Charles Chapman Pugh. All reading and exercises refer to this text, unless otherwise stated. The course will cover most of chapters 1-3 in this text. The specific material covered is as stated in the course catalog:

SCOPE A one semester course providing a rigorous introduction to the calculus of a single variable. The course is designed to introduce the student to the foundations of the calculus necessary for advanced undergraduate and graduate studies in applied mathematics and engineering. Course coverage includes a treatment of the structure of the real number system, sequences, continuous functions, and differentiation.

A recommended supplemental text is Principles of Mathematical Analysis by Walter Rudin. This is a classic text upon which most other real analysis texts are based.

Course Requirements

The emphasis of this course is your ability to read, understand, and communicate mathematics, so most of the course grade will be based on homework assignments. There will also be three WPR's and two course projects.

The course will contain 1000 total points, divided as follows:

Graded Event(s) Date (if any) Points Percent
WPR I 12-Feb 100 10%
Mini-Project 13-Mar 50 5%
WPR II 14-Apr 150 15%
WPR III 2-May 100 10%
Project 15-May 100 10%
TEE 19-23 May 250 25%

Click here for my expectations on written work, including documentation, etc. Final grades will be assigned in accordance with USMA D/Math policies.

Homework

The graded homework will be the primary means of evaluating and providing feedback on your progress. I expect you to complete all of the problems listed on the syllabus, and more if necessary for understanding. I will collect the homework and make every effort to provide prompt feedback on your work. Whenever you write up a problem, always (i) rewrite the problem, in your own words, (ii) state clearly the givens in the problem, and the goal, and (iii) use formal mathematical language. See also the page on how to write proofs.

Homework will be collected about once a week. See the expectations on homework and grading.

I recommend that you work on homework in groups, and come in for AI whenever necessary. You are strongly encouraged to ask questions on the discussion board. I will post answers there.

WPRs & TEE

There will be three midterms and a final. Each exam will cover a single chapter (or portion thereof). Problems on exams will be similar to those on homework assignments. For each exam, the class will collaboratively develop a glossary which they can use for the exam. The final will be comprehensive.

Projects

The projects will involve researching and writing about a topic of your choosing. The projects will be posted on the course wiki, and reviewed by fellow classmates.

Online Participation

In the Web2.0 world, more and more of reading, writing, and communicating mathematics occurs online. A major component of the course will be the course wiki at http://usma387.wikidot.com, which will provide an opportunity for you (as students) to collaborate together, and for me (the instructor) to make answers to your questions visible to all.

During the first week of class, take the following steps:

1. Go to the course wiki and apply for membership. You will need a Wikidot membership (please use your real name!)
2. Post some information about yourself (this will not be visible to anyone outside the class).

During the semester, you will use the wiki to:

2. Collaboratively write a glossary of definitions for use on exams.
3. Post course projects.

There are substantial resources available online to help you learn the Wikidot markup, which is quite simple. You will also need to learn the basics of TeX, the standard language for typesetting in the mathematics community.

Course Policies

• My door is always open for questions, help, or other matters. Feel free to stop by anytime.
• Schedule an appointment if you want to be sure I'm in my office.
• Feedback is extremely valuable to me. Please let me know if there's anything that irks you (or that you really like). I will also solicit feedback several times during the semester. Your honesty is always appreciated.
• It is extremely important to keep up with the work in this course. Late homework will be subject to a 10% per day deduction.
• Makeups for exams and other activities must be arranged at least a week in advance.
• Document all help received on homework and projects in accordance with USMA policies.
• WPR's and the TEE will be closed book, closed notes with the exception of the online glossary. No calculators will be permitted.

Real analysis abounds with the surprises and beauty that makes mathematics so inviting. Most students find that it is not an easy course… you should expect to put a lot into this course, but I expect that you will get a lot out of it as well.

This syllabus will be filled out more as the semester progresses. Check back often for updates and extras.

I expect the reading to be done prior to coming to class, and the exercises to be started.

Lesson 1

On day 1, we will look at the "Big Picture" of mathematics, and how real analysis fits in. We'll also look at some of the surprising results in store for us this semester, and one of my favorite proofs.

Think About: What are the major themes of mathematics? major techniques?

Lesson Date Section Reading Assignments Highlights & Extras
1 14-Jan 1.1: Preliminaries pp. 1-10 Ch1: 1,3,5 my favorite proof

Block I: Real Numbers

In the first block, we will review proof techniques and learn about how to construct the real numbers starting with the natural numbers 0,1,2,… We will also take a look at how the reals are used to construct Euclidean space.

Think About: How many numbers are there? Are there more real numbers than rational numbers? Why do you think so? What does continuity really mean? What are some real-world situations in which the notion of continuity (or discontinuity) appears?

Lesson Date Section Reading Assignments Highlights & Extras
2 16-Jan 1.1: Preliminaries pp. 1-10 Ch1: 6,7 Hilbert and mathematical rigor
(Skill: proof writing)
3 18-Jan 1.2: Cuts pp. 10-17 Ch1: 9,10,11,12,14,15,17 construction of the Reals
4 23-Jan 1.2: Cuts pp. 17-21 [Turn-in: 1,3,5,6,7,9!,10!] Dedekind
5 25-Jan 1.2: Cuts (Fun Math: geometric spaces)
6 29-Jan 1.3: Euclidean Space pp. 21-27 Ch1: 18,26
[Turn-in: 11,14!,15!,18]
7 31-Jan 1.6: Continuity pp. 36-39 Ch1: 19,38
8 4-Feb 1.6: Continuity [Turn-in: these problems] points of discontinuity
9 6-Feb 1.4: Cardinality pp. 28-32 Ch1: 25,32 Cantor & the uncountability of the Reals
10 8-Feb Applications/Review [Discuss: 19,38] (Application: cake cutting)
The Fair Division Calculator
11 12-Feb WPR I about WPR I [Turn-in: 19,38]

Block II: Sets & Sequences (Topology)

In this block, we examine series and sequences of real numbers, and answer the question of when a limit of these numbers exists. We also look at what happens when sets are equipped with a notion of distance. Much of the required terminology comes from the area of Topology, which is essentially a mathematical study of the "closeness" of points.

Think About: What makes a sequence of points converge? Is it possible for a sequence to converge to more than one number? What if you're allowed to remove any terms you want from the sequence? What if you're allowed to rearrange the sequence? What about a series, in which you look at sums of the terms in a sequence?

Lesson Date Section Reading Assignments Extras
12 14-Feb 2.1: Metric Spaces
(sequences and subsequences)
pp. 51-54 Ch2: 26* Exams returned and discussed
sequences/series
13 19-Feb 2.1: Metric Spaces
(metrics; continuity)
pp. 51-56 Ch2: 15,24,82 Mini-project assignments
metric spaces
14 21-Feb 2.1: Metric Spaces
(open and closed sets)
skip Homeomorphism
pp. 58-64 Ch2: 3*,5*,13* Leonhard_Euler and the Konigsberg Bridges
15 25-Feb 2.1: Metric Spaces
(continuity; inheritance)
pp. 55-56,64-69 Ch2: 6,17
[Turn-in: these problems]
(Fun Math: donuts & coffee cups)
16 27-Feb 2.1: Metric Spaces
(product metrics; Cauchy sequences)
pp. 71-75 Ch2: 27 the taxicab & Facebook metrics
17 29-Feb 2.1: Boundedness p. 75 Ch2: 29*,30*
Paper selection for projects
contractions & fixed points
(Fun Math: The Brouwer fixed point theorem)
(Skill: counterexamples)
18 4-Mar 2.2: Compactness I pp. 76 [Turn-in: these problems] Frechet and the importance of compactness
19 6-Mar 2.2: Compactness II pp. 76-82 Weierstrass; the Bolzano-Weierstrass & Heine-Borel Theorems
20 10-Mar 2.5: The Cantor Set pp. 95-99 [Turn-in: these problems] Cantor sets
21 12-Mar 2.5: The Cantor Set pp. 95-99 Mini-projects due by 13-Mar fractional dimensions
(Fun Math: weird spaces)
Spring Break

After Spring break, we will turn to the question: why is compactness SO important? We will also look at the most significant theorems which require the condition of compactness in a very fundamental way. Finally, we'll look at an alternate definition of compactness… which looks very different but turns out to be equivalent.

Lesson Date Section Reading Assignments Extras
22 25-Mar Review of 2.1-2.2
2.3: Connectedness
pp. 82-87 Bernard_Bolzano
the topologist's sine curve
23 27-Mar Applications of Compactness I pp. 79-82 (Application: optimization)
24 31-Mar 2.4: Coverings pp. 88-95 [Turn-in: these problems] space-filling curves
25 2-Apr Applications of Compactness II (Application: approximations)
26 4-Apr 2.4: Coverings (Application: sensor coverings)
27 8-Apr Applications [First three glossary submissions due] (Application: voting)
28 10-Apr Review [Turn-in: these problems as part of WPR grade]
29 14-Apr WPR II about WPR II All glossary entries must be completed by 1600 on 13-Apr

Block III: Functions, Continuity, and Differentiation

In this final block, we will take a look at functions, continuity, and differentiation. Now that the concept of limit has been made rigorous, the concept of a derivative can be as well!

Think About: What conditions must you add to make a continuous function differentiable? What is "lost" (or "gained") at points where a function is not differentiable?

Lesson Date Section Reading Assignment Highlights & Extras
30 16-Apr 3.1: Definition of the Derivative
Mean Value Theorem
pp. 139-143
31 18-Apr Course Drop
32 22-Apr 3.1: L'Hopital's Rule and Discontinuities pp. 143-146 [Turn-in: these problems]
33 24-Apr 3.1: Power Series and Taylor's Theorem pp. 149-151
34 28-Apr 3.1: Power Series and Taylor's Theorem pp. 149-151
35 30-Apr Review
36 2-May WPR III about WPR III All glossary entries must be completed by evening on 1-May

Course End

The last few weeks will be spent on review and topics of your choosing. You will get the chance to research, write, and speak about a topic related to analysis.

Lesson Date Section Reading Assignments Highlights & Extras
37 6-May Radius of Convergence/Power Series 3.1,3.3 [Turn-in: these problems as part of WPR grade]
Projects Day (8-May)
38 9-May Project Drop/IPR's
39 13-May Project Presentations I
40 15-May Project Presentations II
TEE 20 May TEE Comprehensive Th319 0735-1005 Good luck!

This document explains expectations for homework and other written work.

• It is an expectation that you complete all assigned reading in the course.
• You don't need to understand absolutely everything, but you must be prepared to ask questions about it.
• Quizzes may be given on the reading.

Homework

• I expect you to write out the problem to be solved for each homework problem.
• Proofs on homework should be written out in detail. (I will let you know if you're writing too much!) See the guidelines on how to write proofs.
• You may work in groups, but each person should write up their solution separately.
• Homework will be collected once a week, on the Monday or Tuesday following a weekend. The schedule may be altered before an exam.
• Include a cover sheet as with USMA policy. Handwritten acknowledgment of those who worked with you is okay.
• I may ask you to resubmit homework assignments.
• Grades will be based on both completeness and clarity of proof.

Glossaries

• Each student is required to add a minimum of two entries to each glossary.
• Entries should be in your own words and not copy/pasted from elsewhere, or typed verbatim from the text.
• Students may work together to divide terms and theorems into "Very Important" and "Somewhat Important" categories.

Exams

• Glossaries will be available for use on the in-class portion of exams. No other resources will be permitted.
• There is no need to rewrite the problem, as is expected on the homework.
• Proofs should follow a logical style, as on the homework.
• Grades will be based on both completeness and clarity of proof.

Projects

The course projects will be written for a broader audience, and be graded more closely than the homeworks. In addition, you will receive feedback from other students in the course regarding your projects. Work will be presented on the course website, as well as in class. More details will be given at a future date.

Guidelines for Writing Proofs

Page 40 of the text gives an example of the proof technique that you are expected to use. My expectations for homework are as follows:

1. When you begin a problem
• always write out the problem statement (in your own words).
2. When you begin writing the proof
• before the proof comes, write and underline "Proof:"
• always define your variables, and state what set they are contained in (e.g. $x\in\mathbb{R}$ or $x\in A$ or something of the sort)
• begin by writing out the goal of the proof. I use "WTS:", meaning "Want to show"
3. If you are breaking a problem into cases:
• write and underline "Case 1:", "Case 2:", etc.
• at the end of checking a particular case, draw a check mark $\checkmark$
4. If you are using a proof by contradiction
• begin by writing "BWOC, suppose that…", where BWOC means "By way of contradiction"
• when a false statement is reached, write "This is a contradiction, therefore…" or something similar.
5. At the end of the proof
• draw a small box $\blacksquare$, lightning bolt, "Q.E.D.", "HOOAH", or some symbol of your choosing to indicate the end of the proof.

Proofs by contradiction can be very useful. Here is an example:

Theorem: There are an infinite number of prime numbers.

Proof:
BWOC, suppose there were only $n$ primes, call them $\{p_1,p_2,\ldots,p_n\}$. That would mean that every positive integer can be expressed as a product of elements in this set. We will establish a contradiction by finding an integer for which this is not true.

Let $q=p_1p_2\cdots p_n$, which divides every prime. However, $q+1=p_1p_2\cdots p_n+1$ does not divide any of the primes, since it has remainder 1 when divided by any of them. Therefore, $q+1$ cannot be expressed as a product of elements in the set $\{p_1,\ldots,p_n\}$, contradicting the assumption that there were only $n$ primes.

Therefore, no finite set of integers can contain all the prime numbers, and the set of prime numbers is infinite. $\blacksquare$

Epsilon-Delta Proofs

These are extremely important in this course. Typically, the goal is to find $\delta$ such that if some condition involving $\delta$ is true, then some condition involving $\epsilon$ is true.

For example, to show a function $f$ is continuous at a point $x$, one must show that

(1)
\begin{align} \text{there exists a } \delta>0 \text{ such that for all $u$ satisfying } |x-u|<\delta, |f(x)-f(u)|<\epsilon. \end{align}

The proof that some function is continuous will typically go like this:

Proof:
Let $x\in\mathbb{R}$ and let $\epsilon>0$.
WTS there exists $\delta>0$ such that if $|x-u|<\delta$ then $|f(x)-f(u)|<\epsilon$.

Let $\delta=something$.
Then $|f(x)-f(u)|= \quad\cdots\qquad\cdots\qquad\cdots\qquad < \epsilon$.
$\blacksquare$

Note that if you are just showing $f(x)$ is continuous on a particular domain, or at a particular point, you would replace $x\in\mathbb{R}$ with something more appropriate.

There are two pieces which must be filled in. First, you have to figure out what $something$ to use for $\delta$. This may be tough to figure out, but usually can be discovered by some scratchpad pencil work. Second, you have to figure out how to put together a string of inequalities starting at $|f(x)-f(u)|$ and ending up at $\epsilon$. This also involves some scratchpad work, and frequently a trick or two.

Counterexamples

To show that a statement is false, it suffices to find a counterexample. This is a sort of "unproof", meaning a way to show that a statement is not absolutely true.

This page contains notation that has been defined in the course. Feel free to add new entries.

To edit this page, click the button that says "edit" at the bottom. Follow the examples shown to write the entry.

Example
Use colons to define the term. Include spaces between the colons and the surrounding words. Use double stars around the title of the term to make it bold. Use [[$and$]] to write mathematical terms such as $\langle A,B\rangle$.

This example is produced by typing

: **Example** : Use colons to define the term. Include spaces between the colons and the surrounding words. Use [[$and$]] to write mathematical terms such as [[$\langle A,B\rangle$]].

See how to edit pages for more help.

Notation and Abbreviations

Proof notation

The following notation is commonly used in proofs:

• $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$, and $\mathbb{R}$ for the sets of natural numbers, integers, rationals, and reals, respectively. $\mathbb{C}$ is used for complex numbers;
• $\in$, as in $x\in A$ to indicate that an element is contained in a set [you should not use a curly epsilon $\varepsilon$];
• $\Rightarrow$ meaning implies, and $\Leftarrow$ meaning only if;
• $\Leftrightarrow$ meaning if and only if or is equivalent to;
• $\{n\in\mathbb{Z}: n \text{ is even}\}$ or $\{n\in\mathbb{Z}| n \text{ is even}\}$to indicate a set constructed of all elements with a given property. Here the colon $:$ is read as "such that";
• $f:A\to B$ to indicate a particular function's domain and range, read as "$f$ takes $A$ to $B$";

The following abbreviations are also commonly used:

• BWOC: by way of contradiction;
• WTS: want to show;
• QED: quad erat demonstratum or that which was to be shown;
• IFF: if and only if;
• WLOG: without loss of generality.

The following notations are less commonly used; I do not advise you to use them very often in proofs:

• $\forall$ meaning "for each" or "for all";
• $\exists$ meaning "there exists", and $\nexists$ meaning "there does not exist";
• $\therefore$ meaning "therefore";
• $\ni$ or $|$ or $:$ meaning "such that" ($|$ and $:$ are usually used only in set notation);

More Notation

These notations are specific to particular mathematical structures (e.g. logic, equivalence relations)

Equivalence relations
$\sim$ means equivalent;
Logic
$\wedge$ means "and", $\vee$ means "or", and $\lnot$ means "not";
Set Theory
$\emptyset$ represents the empty set, $A\cap B$ represents the intersection of sets, $A\cup B$ represents the union of sets; $A^c$ represents the complement of a set; $A+A'$ represents the set of all elements which can be written as the sum of an element in $A$ and an element in $A'$; $A*A'$, $\frac{1}{A}$, etc. are defined similarly.
Cuts
$A|B$ indicates a cut.
Inner products
$\langle \vec x,\vec y\rangle$ and $\vec x \cdot \vec y$ both indicate the dot product or inner product of two vectors
Euclidean space
$(a,b)$ indicates the open interval $\{x:a<x<b\}$ of the real numbers and $[a,b]$ represents the closed interval $\{x:a\leq x\leq b\}$; $\mathbb{R}^n$ represents the set of n-vectors of the form $(x_1,\ldots,x_n)$; the Cartesian product $A\times B$ of sets $A\subset\mathbb{R}$ and $B\subset\mathbb{R}$ is the set of vectors $\{(a,b):a\in A,b\in B\}$.

To edit a page, click the edit button at the bottom of the page. This will open an editor with a toolbar palette with options. When you are done, click save. If you are composing a longer document, hit save & continue frequently!

If you have questions beyond what is answered here, post a question on the discussion board.

Tips on editing pages

• You may wish to keep a window open with the Wiki Syntax description the first few times you edit a page.
• To create a link to a page, use the syntax: [[[page name]]] or [[[page name | text to display]]]. It's okay if the page doesn't exist yet… follow the link to create the page.
• On any page, you should be able to click the + options button at the bottom of the page and view source to see examples of how certain things are done

Tips on TeX and Mathematical Notation

What is TeX??
$\TeX$ (pronounced "tech"), and $\LaTeX$ (pronounced "lay-tech"), are typesetting systems which are especially suited to mathematical language. They require memorizing (or looking up) "commands" for creating certain kinds of characters, but the output is much better than Equation Editor!

Examples

$\TeX$ is fairly intuitive, and using it requires you to know a few basic commands. Here are some examples of how it's done. In general you write out [[$whatever$]], where "whatever" is the formula you wish to put on the page.

Type Result Explanation
[[$f(x)$]] $f(x)$
[[$\sum_n a_n x^n$]] $\sum_n a_n x^n$ Use an underscore _ for subscript and a caret ^ for superscript.
for all [[$x\in\mathbb{Z}$]] for all $x\in\mathbb{Z}$ Use \in for $\in$, and use \mathbb for the "blackboard bold" font.
[[$\langle \vec{x},\vec{y}\rangle$]] $\langle \vec{x},\vec{y}\rangle$ The \vec{x} command places a vector over the character x
[[$\lim_{h\to 0} \frac{f(x+h)-f(x)}{h}$]] $\lim_{h\to 0} \frac{f(x+h)-f(x)}{h}$ The \frac{num}{denom} produces a fraction $\frac{num}{denom}$. An arrow $\to$ is produced using \to. For the left arrow $\leftarrow$, use \leftarrow.
[[$\{a\in\mathbb{Q}:a^2\leq 2\}$]] $\{a\in\mathbb{Q}:a^2\leq 2\}$ Braces are produced in TeX by preceding them with a backslash: \{ and \}

More symbols

The following table shows more examples of commands which may be useful and their output.

Command Result Command Result Command Result Command Result
\mathbb{Z} $\mathbb{Z}$ \mathbb{Q} $\mathbb{Q}$ \mathbb{N} $\mathbb{N}$ \mathbb{R} $\mathbb{R}$
\vec{x} $\vec{x}$ \mathbf{x} $\mathbf{x}$ \mathsf{x} $\mathsf{x}$ \mathcal{X} $\mathcal{X}$
x_n $x_n$ x^n $x^n$ \frac{x}{y} $\frac{x}{y}$ \binom{x}{y} $\binom{x}{y}$
\sqrt{x} $\sqrt{x}$ \sqrt[3]{x} $\sqrt[3]{x}$ \sqrt[n]{x} $\sqrt[n]{x}$
\in $\in$ \forall $\forall$ \exists $\exists$ \nexists $\nexists$
\cup $\cup$ \cap $\cap$ \subset $\subset$ \supset $\supset$
\to $\to$ \leftarrow $\leftarrow$ \Rightarrow $\Rightarrow$ \Leftarrow $\Leftarrow$
\longrightarrow $\longrightarrow$ \longleftarrow $\longleftarrow$ \Leftrightarrow $\Leftrightarrow$ \Longleftrightarrow $\Longleftrightarrow$
\neq $\neq$ \geq $\geq$ \leq $\leq$ \approx $\approx$
\langle $\langle$ \rangle $\rangle$ |\vec{x}| $|\vec{x}|$ | |\vec{x}| | $| |\vec{x}| |$
\{a\} $\{a\}$ a\cdot b $a\cdot b$ (x_1,x_2,\ldots,x_n) $(x_1,x_2,\ldots,x_n)$ \cdots $\cdots$
\infty $\infty$ \overset{\circ}{S} $\overset{\circ}{S}$ \overline{S} $\overline{S}$

For more commands, see below for helpful webpages on $\TeX$.