Real Analysis I.
    Spring 2013

    Professor Francis Su
    Office Hours: WED 2 - 4pm, or by appointment. My office is: Olin 1269.
    e-mail: [my last name] @

    Graders/Tutors: Rosalie Carlson, Liz Sarapata, Eric Stucky
    Tutoring hours:
    TUE and WED 7-9pm
    Sprague 1st floor Learning Studio
    e-mail: [firstname_lastname] @ (skip apostrophes).

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:

Required Text: Walter Rudin, Principles of Mathematical Analysis, McGraw-Hill. We will cover Chapters 1 through 5. 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 (Olin 1269) by 5:15pm on Thursdays. 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:

Late homeworks can only be accepted by special permission. Please ask at least 24 hours in advance. The lowest homework assignment will be dropped. Please follow the HMC Mathematics Department format for homework, online at

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.

Midterms and Grading: There will be three mid-terms:

  1. Handed out: Mon Feb 25, due Thu Feb 28.
  2. Handed out: Mon Apr 1, due Thu Apr 4.
  3. Thu May 9 for seniors and Monday 9am-noon May 13 for everyone else.
Each of these and your homework average will count 25% of your course grade.

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. Note that some of the homework problems in this course have been assigned in prevous semesters. Copying work from published solutions (or solutions of past students) is a violation of the HMC Honor Code and will be handled accordingly.

Taped YouTube Lectures:

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! (The taped classes were 75 minutes long, so the lecture numbers will not line up with this year's MWF 50 minute classes.)

Real Analysis Lectures, Spring 2010.

Homeworks, due Thursdays at 5:15pm, in box outside my office (Olin 1269).
All HW's refer Rudin's Principles of Mathematical Analysis.

  • HW #0. Due Thursday Jan 24 at 5:15pm. Read this handout on good mathematical writing and turn in brief answers to these questions. Keep in mind the handout and the homework format as you write up your answers.
    • Directly from the handout reading:
      • 1. What is a good rule of thumb for what you should assume of your audience as you write your homework sets?
      • 2. Is blackboard writing formal or informal writing?
      • 3. Do you see why the proof by contradiction on page 3 is not really a proof by contradiction?
      • 4. Name 3 things a lazy writer would do that a good writer wouldn't.
      • 5. What's the difference in meaning between these three phrases?
          "Let A=12."
          "So A=12."
    • Now examine Section 1.1 of Rudin, showing that there is no rational p that satisfied p2=2.
      • 6. 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?
      • 7. What would you change about his presentation if you were writing for a high school audience? Give a specific example.

  • HW #1. Due Jan 31. Read Chapter 1 in its entirety. Do not worry about understanding everything, just read for the big ideas. Turn in the following problems:

    • Problem A. In no more than four sentences, describe the main themes and concepts of Chapter 1. The first sentence or two should be at a level that your parent could understand even if they never went to college. The other sentences should be understandable by any college student.

    • Problem B. Recall that in class, we defined a rational number 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)~(m,n) and (m,n)~(a,b), then (p,q)~(a,b).

    • Problem C. We defined addition of rational numbers in terms of representatives: a/b + c/d = [ad+bc]/[bd]. Show that the addition of rational numbers is well-defined.

    • Problem D. Define a multiplication of rational numbers, and show this multiplication is well-defined.

    • Do also Chapter 1 ( 1, 2, 3a ).

    Follow this homework format as well as the guidelines for good mathematical writing.

p.s. Be sure to subscribe to math-131-l (using if you are not already getting e-mails from the list.

  • HW #2. Due Feb 7. 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 is 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.]

If you want to do rewrites, see this guide.

p.s. Be sure to subscribe to math-131-l (using if you are not already getting e-mails from the list.

Rewrites for HW#1 are also due. If you want to do rewrites, see this guide.

  • HW #4. Due Feb 21. A problem marked "R" means read, but do not do the problem.

      Do Chapter 2 ( 2, R3, 4, 5, R6, R7, 8, 11[exclude d2] ).
      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.]

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.

  • EXAM 1. Available Monday Feb 25 (afternoon). Though test is due FRIDAY Mar 1 at 2:30pm.

Rewrites for HW#3 are still due Thursday 2/28.

  • HW #5. Due Mar 7. A problem marked "R" means read, but do not do the problem. There are no re-writes on this homework and forward.

      Do Chapter 2 ( 9ab, 9cd, 9ef, R10, 12, R14, 16, 22, R23 ).
      In Problem 14, give an example of a cover that is not a nested collection of open sets.

Rewrites for HW#4 are also due.

  • HW #6. Due Mar 14. 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, 17[countable?, dense?], 17[compact?, perfect?], 18, 19ab, 19cd, 24, R25).
      (R25 depends on R23 from last week's reading.)

    Other helpful facts: recall that rational numbers have decimal expansions that either terminate or eventually repeat. I encourage you to discuss these problems with others in the class!

No further rewrites accepted.

Spring Break Week

  • HW #7. Due Mar 28. A problem marked "R" means read, but do not do the problem.
      Do Chapter 2 ( 20, R26 ) Chapter 3 ( 1, 3, R4, R16, 20, 23, R24, R25 )
      Hint on 3.3: can you show the sequence is increasing? Induction may be of help here.

    Please have a look at these problems before spring break. Have a great break!

  • EXAM 2. Due Thu Apr 4.

    You may wish to review your notes in preparation for the midterm.

  • HW #8. Due Thu Apr 11. A problem marked "R" means read, but do not do the problem.
      Do Chapter 3 ( 4, 7, 8, R9, 10, 16a, 16bc, R17 )

  • HW #9. Due Thur Apr 18. A problem marked "R" means read, but do not do the problem.
      Chapter 4 ( 1, 2, 3, 4, R6, 8 ) and this
      Problem M. Carefully write out the details to the proof of Theorem 4.10(part a).
    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 R2 (where the graph lives) is the usual Euclidean metric.
Don't forget the Moody Lecture on Friday April 19 at 7pm: "The Mathematics of Doodling". It's related to analysis, but will be accessible to all! Invite your friends! Watch your e-mail for how to earn extra credit.

  • HW #10. Due Apr 25. A problem marked "R" means read, but do not do the problem.
      Do Chapter 4 ( 9*, 10, 11**, 12, 14, R16, 18, R19 ).
      *The definition of diameter can be found on page 52.
      **You do NOT have do the 2nd part of Problem 4.11, where it says "Use this result to give..."
    Please come see me or the tutors to discuss these problems!

  • HW #11. Due May 2.
      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, 4, 13ab*, R13cdefg, R25abd, 25c ) and Chapter 7 ( 1, R2, R3 ).
      *In Problem 5.13, there is a typo (in some editions of the book): the xa should say |x|a.
The Final Exam will NOT be a take-home. See details in your e-mail.