Math 131: Real Analysis I

Real Analysis I.
Spring 2008
http://www.math.hmc.edu/~su/math131/

Professor Francis Su
Office Hours: Olin 1269, WED 2:30 - 4pm or by appointment.
e-mail: [my last name] @ math.hmc.edu

Graders/Tutors: Sarah Fletcher, Hendrik Orem, Mutiara (Tia) Sondjaja, Andres Perez
8 - 9:30 pm, TUE and WED evenings, Platt Living Room
e-mail: [firstname_lastname] @ hmc.edu

This course is a rigorous analysis of the real numbers. 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 of real analysis.
Learn to read and write rigorous proofs.
Learn good mathematical writing skills and style.

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 2pm 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, which will work as follows:

Turn in the homework by 2pm 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 by 2pm 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 not be done for assignments due after Spring Break.

Late homeworks will not be accepted, but the lowest homework assignment will be dropped. Please follow the HMC Mathematics Department format guidelines for homework, online at http://www.math.hmc.edu/teaching/homework/.

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

Handed out: Mon Feb 25, due Thu Feb 28.
Handed out: Mon Apr 7, due Thu Apr 10.
Handed out: Wed Apr 30, due Thu Apr 8.

Each of these and your homework average will count 1/4 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 dealt with accordingly.

Homeworks, due Thursdays at 2pm, in box outside my office (Olin 1269).

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. We also 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 C. Define a multiplication of rational numbers, and show this multiplication is well-defined.
Do also Chapter 1 ( 1, 2, 3a ).

In all homework, remember to concentrate on good writing. Some of these handouts may help, even if the handouts are aimed at the calculus student. In particular, please read the handout on "Flow".

HW #2. Due Feb 7. A problem marked "R" means read, but do not do the problem.
Do Chapter 1 ( 3d, 4, 5, 6, 8, R9 ).

HW #3. Due Feb 14. Do the problems on this handout. If you would like the LaTeX code, it is here .

HW #4. Due Feb 21. A problem marked "R" means read, but do not do the problem.
Do Chapter 1 ( R10, 12, 13, 15 ) and
Do Chapter 2 ( R1, 2, R3, 4, 5 ).
Comments: In 1(15) you should explain why.

EXAM 1. Handed out Monday Feb 25. Due FRIDAY Feb 29 at 2pm. Slip under my office door. (Re-writes for HW 3 still due Thursday 2/28 at 2pm.)

HW #5. Due Mar 6. A problem marked "R" means read, but do not do the problem.
Do Chapter 2 ( R6, 7, 8, 9, R10, 11[exclude d2], 22, R23 ).

HW #6. Due Mar 13. A problem marked "R" means read, but do not do the problem.
Do Chapter 2 ( 12, 14, R15, 16, 17, 18, R23, 25 ).
In Problem 14, give an example of a cover that is not a nested collection of open sets.
Other helpful facts: recall that rational numbers have decimal expansions that either terminate or repeat. Remember that to show a set is compact you must start with an arbitary open cover and show how to produce a finite subcover. I encourage you to discuss these problems with others in the class.

HW #7. Due Mar 27. A problem marked "R" means read, but do not do the problem.
Do Chapter 2 ( 19, 20, R24, R26 ) Chapter 3 ( 1 )
It's shorter than usual. Please have a look at it before the weekend. Have a happy spring break!

HW #8. Due Apr 3. A problem marked "R" means read, but do not do the problem.
Do Chapter 3 ( 4, R16, 20, 23, 24abc, R24de, R25 )

EXAM 2. Handed out Monday Apr 7, due Thursday Apr 10 at 4pm. You may wish to review your notes in preparation for the midterm.

HW #9. Due Apr 17. A problem marked "R" means read, but do not do the problem.
Do Chapter 3 ( 8, R9 ) Chapter 4 ( 1, 2, 3, 4, 6 )
On 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.

HW #10. Due Apr 24. A problem marked "R" means read, but do not do the problem.
Do Chapter 4 ( 8, 10, 12, 14, 16, R17, 18, R19 ).

HW #11. Due May 1. Absolutely NO homeworks will be accepted after this time.
Do:
Problem D. 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 ( R1, 4, 13ab, R13cdefg, 25abcd )
and Chapter 7 ( 1, R2, R3 ).
Optional: for the last exam, I invite you to submit a possible test question for the exam (true/false, short answer, or regular problem). Write up the problem and a nicely written solution (no longer than a page). Turn it in directly to me no later than Monday April 28, and I may use it on the exam!
Last exam will be available Tue May 6.