### Real Analysis I – Spring 2016

Office Hours: TUE 11am–noon and 1:30–2:30pm, or by appointment.

Graders/Tutors: Patrick Tierney, Jared Tramontano, Sam Miller, Hope Yu, Kat Dover
Tutoring Hours: TUE and WED 6–8pm.
Sprague Library, 1st floor.

This course is a rigorous analysis of the real numbers, as well as an introduction to writing and communicating mathematics well. Topics will include: construction of the real numbers, fields, complex numbers, topology of the reals, metric spaces, careful treatment of sequences and series, functions of real numbers, continuity, compactness, connectedness, differentiation, and the mean value theorem, with an introduction to sequences of functions. It is the first course in the analysis sequence, which continues in Real Analysis II.

## Goals of the course

• Learn the content and techniques of real analysis, so that you can creatively solve problems you have never seen before.
• Learn to read and write rigorous proofs, so that you can convincingly defend your reasoning.
• Learn good mathematical writing skills and style, so that you can communicate your ideas effectively.

This class is about the exciting challenge of wrestling with big ideas. I believe everyone in the class is fully capable of mastering this material. Questions are valued, especially simple ones because they can lead to profound ideas. Exploration is encouraged, especially risk-taking in trying out things that may not work, because they can lead to further areas of exploration. I expect all of us to be welcoming of the questions and explorations of others.

## Required Text

Walter Rudin, Principles of Mathematical Analysis, McGraw-Hill. We will cover Chapters 1 through 5, and part of Chapter 7. There are also many other books on analysis that you may wish to consult in the library, around the QA300 area.

## Homeworks, and Re-Writes

Due at my office (Shan 3416) by 1:15 pm on Thursdays. Please follow the HMC Mathematics Department format for homework. Because I want you to learn from the feedback you get on your homework, as well as improve your writing skills, I will use a system of (optional) re-writes for the first few assignments, which will work as follows:

• Turn in the homework on the due date.
• The homework will be graded and returned to you within one week.
• If you are not satisfied with the grade you received on the homework, you have the option of re-doing any question(s) you wish, and submitting the re-written version together with the previously graded version. (You may only re-write a question if you made a serious attempt at it on the first version.)
• If you choose to do a re-write, it is due at my office two weeks after the original due date of the assignment. Your re-write will be graded, with particular attention to whether you adopted the graders’ suggestions, and new grades will be assigned for rewritten questions. Your grade for a rewritten question will always go up or stay the same; it will never go down. Rewrites will only be accepted for Homeworks 1 through 3.

### Late Homeworks

Late Homework can be accepted (with penalty) by special permission. Please ask at least 24 hours in advance

### LaTeX

Some of you may find LaTeX helpful in typesetting your homework. If you’d like to learn LaTeX, or have questions about it, you can visit the CCMS Software Lab.

There will be three exams:

1. Midterm 1: Part in-class, part take-home. In class portion: Feb 22. Take-home due Fri Feb 26.
2. Midterm 2: Written exam week of March 30.
3. Final: during the regularly scheduled final period.

Each of these and your homework average will count 25% of your course grade. The lowest homework assignment will be dropped. It is helpful to remember that course grades are just intended to assess what you have learned. But they are a not a reflection of your potential ability or your dignity!

## Honor Code

The HMC Honor Code applies in all matters of conduct concerning this course. Though cooperation on homework assignments is encouraged, you are expected to write up all your solutions individually. Thus copying is prohibited, and you should understand your solutions well enough to write them up yourself. It is appropriate to acknowledge the assistance of others; if you work with others on a homework question, please write their names in the margin. Part of the fun of this course is the struggle, as well as the joy of discovering a solution for yourself. Please note: using solutions found online or solutions of previous students will be regarded as a violation of the HMC Honor Code and will be handled accordingly. I encourage you instead to talk to me or the tutors or each other!

These lectures were taped in 2010, and although the lectures I give this year may not be identical, they will be close enough that you may find it valuable to use them for review. Or, better yet, watch them before the class lecture, and then during class you can ask questions! I do not encourage using these lectures as a substitute for class, however, since we will be doing slightly different things and interactions with me and other students will be critical for your learning.

Real Analysis Lectures, Spring 2010.

## Homeworks

All HW’s refer Rudin’s Principles of Mathematical Analysis.

Read this article by Carol Dweck and my handout on good mathematical writing. Fill out this survey. Then turn in brief answers to these questions. Keep in mind the handout and the homework format as you write up your answers.

From the article by Carol Dweck:

1. When faced with a mathematical challenge, describe 3 ways a person with a growth mindset would respond differently than a person with a fixed mindset.

Directly from the handout on good writing:

1. What is a good rule of thumb for what you should assume of your audience as you write your homework sets?
2. Do you see why the proof by contradiction on page 3 is not really a proof by contradiction?
3. Name 3 things a lazy writer would do that a good writer wouldn’t.
4. What’s the difference in meaning between these three phrases?
$$Let A=12.$$
$$So A=12.$$
$$A=12.$$

Now examine Section 1.1 of Rudin, showing that there is no rational $$p$$ that satisfied $$p^{2}=2$$

1. There are many places in his proof where he could have used symbols to express his ideas, but he does not. (e.g., “Let $$A$$ be the set of all positive rationals $$p$$ such that…”) Why do you think he chooses not to use symbols? What would you change about his presentation if you were writing for a high school audience? Give a specific example.

Your homework should be handed in three parts.
Part A should have problems 1 and 2.
Part B should have problems 3 and 4.
Part C should have problems 5 and 6.

• Problem B. Recall that in class, we defined a rational number $$\frac{m}{n}$$ to be an equivalence class of pairs $$(m,n)$$ under an equivalence relation. Check that this equivalence relation is transitive: if $$(p,q)\equiv(m,n)$$ and $$(m,n)\equiv(a,b)$$, then $$(p,q)\equiv(a,b)$$.
• Problem C. We defined addition of rational numbers in terms of representatives: $$\frac{a}{b} + \frac{c}{d} = \frac{ad+bc}{bd}$$. Show that the addition of rational numbers is well-defined.
• Problem D. Define a multiplication of rational numbers (corresponding to the one you are used to), and show this multiplication is well-defined. For problems B, C, D, you may assume you know properties of integers, including the cancellation law of Z. You cannot assume that you know how to divide integers (because that is really multiplication by a rational), nor how to add or multiply rationals (because that is what you are trying to show)!
• Do also Chapter 1 (1, 2, 3a). For these problems in Rudin, you may assume that you know familiar properties of rational numbers that we were proving in Lecture 2.

Your homework should be handed in three parts.
Part A should have problems B and C.
Part B should have problems D and 1.1.
Part C should have problems 1.2 and 1.3a.

A problem marked “R” means read, but do not do the problem.

Chapter 1 (R3bcd, 4, 5, R7, 8, 9, 20) and Problem S. For a real number $$a$$ and non-empty subset of reals $$B$$, define: $$a+B= \{a+b:b \in B\}$$. Show that if $$B$$ is bounded above, then $$sup(a + B) = a + sup(B)$$.

[For 1.20, whenever the proof is EXACTLY THE SAME as in Steps 3 and 4 of pp. 18–19, you do not need to re-write the proof. Just point out that the proof is the same as in Rudin’s. But wherever the proof differs, BE SURE to POINT OUT HOW IT DIFFERS, and VERIFY all new things.]

There will be no rewrites for HW #0.

Your homework should be handed in three parts.
Part A should have problems 4 and 5.
Part B should have problems 8 and 9.
Part C should have problems 20 and S.

Do the problems on this handout. If you would like the LaTeX code, it is here.

Your homework should be handed in three parts.
Part A consists of the first 2 problems in the order above; Part B the second 2 problems, and part C the third 2 problems.

Rewrites for HW #1 are also due. If you want to do rewrites, see this guide. Rewrites should be handed in in parts.

A problem marked “R” means read, but do not do the problem. There are no rewrites on this homework and forward.

Do Chapter 2 (2, 3, Q, T, 4, R6, R7, R8, 11 [exclude the 2nd metric d_2]).

Problem Q. Cantor’s diagonal argument shows that the real numbers are uncountable by showing that any proposed “list” of real numbers must not contain every real number. We know rationals numbers ARE countable, but why can’t Cantor’s diagonal argument be applied to show that every proposed “list” of rationals between $$[0,1]$$ can’t contain every rational number between $$[0,1]$$? What would go wrong with this line of reasoning?

Problem T. Prove that the Principle of Induction implies the Well-Ordering Principle for $$N$$ (the natural numbers).
[Hint: Let $$L(n)$$ be the statement “if $$A$$ is a subset of $$N$$ that contains a number $$<= n$$, then $$A$$ has a least element”.
Now prove $$L(n)$$ holds for all $$n$$ by induction.]

In the homework, you may assume these things:

• Every real number can be represented in a decimal expansion.
• Every rational number has a decimal expansion that eventually repeats (or terminates, which is really repeating zeroes)
• Every polynomial of degree $$n$$ has $$n$$ roots in the complex numbers.

Your homework should be handed in three parts.
Part A consists of 2.2 and 2.3, Part B consists of Problems Q and T, and part C consists of 2.4 and 2.11.

Rewrites for HW #2 are also due. If you want to do rewrites, see this guide. Save a copy since you probably won’t get the homework back in time to study for the exam.

Part of it will be in class Mon Feb 22. The other part will be take-home and available Mon Feb 22, due Fri Feb 26 by midnight.

Rewrites for HW #3 are due. Rewrites should be handed in in parts. No more rewrites after this date.

A problem marked “R” means read, but do not do the problem.

Do Chapter 2 ( 5, 8, 9ab, 9cd, 9ef, R10, 12, R14, R22, R23 ).
In Problem 14, give an example of a cover that is not a nested collection of open sets.

No further re-writes accepted.

A problem marked “R” means read, but do not do the problem. (I’ve split up some problems into pieces to indicate which pieces are worth a similar amount of points, and the graders will grade these parts separately.)

Do Chapter 2 ( R15, 16, 17[countable?, dense?], 17[compact?, perfect?], 18, R19, 22, 24, R25).
(R25 depends on R23 from last week’s reading.)

On problem 2.17, notice a decimal expansion with only 4’s and 7’s cannot contain any 0’s. So there are no terminating decimal expansions. (So, for instance, the set does not contain the number .44 because that’s really .44000…)
On problem 2.18 recall that rational numbers have decimal expansions that either terminate or eventually repeat.
On problem 2.24 you should not appeal to compactness because this problem is one step of the proof that “every subset of $$X$$ has a limit point” implies “$$X$$ is compact”. Instead, follow the hint.

I encourage you to discuss these problems with others in the class!

A problem marked “R” means read, but do not do the problem.

Do Chapter 2 ( 19ab, 19cd, 20[closures], 20[interiors], R26 ) Chapter 3 ( 1, 3, R4, R16, R20, R23[prove the hint], R23[do problem], R24, R25 )
Note on 2.20: it’s asking “is the closure of a connected set also a connected set?” and “is the interior of a connected set also a connected set?”
Note on 3.1: you should assume the sequence is in $$\mathbb{R}$$ or $$\mathbb{C}$$.
Hint on 3.3: can you show the sequence is increasing? Induction may be of help here.
Note on 3.23: you are being asked to show a particular sequence converges. Note that the sequence is a sequence of distances, so they are in $$\mathbb{R}$$.

Written exam available Mon Mar 28, due Fri Apr 1 at 5pm under my office door.

Review class notes and homeworks in preparation for the midterm. The exam will be 2.5 hours long, and you will be allowed a one-page sheet of notes (front and back), that you may have with you during the exam. The sheet must be prepared by you.

A problem marked “R” means read, but do not do the problem.

Do Chapter 3 ( 4, 7, R8, 16, R17, 20, 23, 24b, R24acde )
Hint: on 3.7, the Cauchy-Schwarz inequality may be helpful. On 3.8, is there a theorem in the book that may help?

A problem marked “R” means read, but do not do the problem.

Do Chapter 3 ( 9ab, 9cd, 10, R25 ) Chapter 4 ( 1, R2, 3, 4, R6)

On the reading problem 4.6, you may assume that $$E$$ is a subset of real numbers and $$f$$ is a real-valued function, and the distance in $$R^{2}$$(where the graph lives) is the usual Euclidean metric.

A problem marked “R” means read, but do not do the problem.

Do Chapter 4 (8, R10, 11, 12, 14, R16, 18, R19 ) + Problem W below.

Problem W. Write 3 True/False questions over the most recent material that could appear on the final exam. Include your answer to each of them (true/false) and a brief answer why. In addition to turning it in with your homework, enter it into the google doc that I will send out where everyone can see them.

In 4.11. you do not have to do the 2nd part of the problem where it ways “Use this result to prove…”

Do Problem E. Use the mean value theorem to show that $$e^x$$ is greater than or equal to $$(1+x)$$ for all $$x$$ in $$R$$. (You may assume knowledge of the derivative of $$e^x$$.) and Chapter 5 ( 1, R4, 13ab*, R13cdefg, R25abd, 25c ) and Chapter 7 ( R1, R2, R3 ) and Problems J and K:

Problem F. Look at the proof of Theorem 4.19 in the book, explain and discuss it with others, and re-write the proof from the book in your own words. (Note that it is different than the proof we presented in class.) Indicate whom you discussed this problem with.

Problem G. Look at the totality of all the theorems that involve compactness, discuss them with others, and write a couple of paragraphs explaining its significance of compactness to a student who has never taken analysis. Indicate whom you discussed this problem with.

*In Problem 5.13, there is a typo (in some editions of the book): the $$x^{a}$$ should say $$|x|^{a}$$ and $$x^{-c}$$ should say $$|x|^{-c}$$.
In the problems of Chapter 5, you can assume you know basic facts about the sine function (it’s continuous, it lies between -1 and 1, etc) and basic rules of differentiation (power rule, etc).

### Possible Upcoming Homeworks

Below this line, all homeworks are TENTATIVE. This means they are likely to be assigned, but there is no guarantee that they will until you see them moved to the box ABOVE. I am putting them here in case you want to work ahead!