Instructor: Denis Auroux (auroux@math.harvard.edu)

Office: Science Center 539.
Office hours: Tuesdays and Thursdays, 9:30-11am.
Lectures: Tuesdays and Thursdays, 12:00-1:15pm, Science Center Hall E.

Course assistants:

• Emily Saunders (esaunders@college): section Wednesdays 6-7:15pm in SC 221, office hours Mondays 8-9pm in Leverett Dining Hall (Math Night).
• Valerie Zhang (vzhang@college): section Tuesdays 1:30-2:45pm in SC 222, office hours Fridays 12-2pm in Mather D-Hall.
• Julian Asilis (asilis@college): section Sundays 7:30-8:45pm in SC 221, office hours Mondays 8-10pm in Leverett Dining Hall (Math Night).
• Garrett Brown (garrettbrown@college): section Mondays 1:30-2:45pm in SC 411, office hours Sundays 9:30-11:30am in Winthrop D-Hall.

Textbook: Rudin, Principles of Mathematical Analysis, McGraw-Hill, 3rd edition.

There will be weekly (or near-weekly) homework assignments, a midterm during class period on Tuesday March 12, and a final exam during finals period (Monday May 13). Approximate grading weight: homework 40%, midterm 20%, final 40%.

Syllabus: see below.

Announcements

• (5/16)The final is graded (at last). The the median score was 124 out of 200, the lower quartile was 95, the upper quartile was 142.
• (5/15) Exam grading is still in progress, and will likely take until Friday. I am aware that the final turned out to be more difficult than expected, and that many of you didn't do so well as a result. The grading curve will take this into account. Scores will be posted to Canvas and the score distribution (median and quartiles) will be posted here once grading is complete.
• (5/6) Julian Asilis has provided an updated version of his set of lecture notes (for almost the entire class, missing two days of material).
• (5/1) There will be a review session on Tuesday May 7, 1:15-2:45pm, in Science Center Hall E. Note the unusual time (the room wasn't available at 12).
• (4/23) Some final exam info (including extra practice problems from Rudin) is available.
• (4/11) My office hours on Tuesday 4/16 are cancelled. I will be holding make-up office hours on Monday 4/15 from 1 to 2:30pm.
• (3/16) The midterm has been graded, and scores posted on Canvas. If you are around during spring break, check your email later today for information on when you can get your exam back from me and on the grade distribution. Also, I will be out of town 3/21-3/27 for a conference; my office hours on Tuesday 3/26 are cancelled, and the lecture that day will be taught by one of my colleagues.
• (3/6) Prof Auroux will have extra office hours on Friday March 8, from 3:30pm to 5pm.
• (3/5) Julian Asilis has typed up a short set of lecture notes which you may find helpful in reviewing for the midterm. (Warning: these notes haven't been proofread, may contain errors. If in doubt, check with Rudin).
• (3/3) Midterm information and practice problems are here!
• (2/26) Prof Auroux's office hours on Tuesday March 5 are cancelled.
• (2/23) Garrett Brown's office hours on Sun Feb 24 are cancelled; similarly for Julian Asilis' section and office hours on Sun & Mon Feb 24-25. Garrett and Julian will hold makeup office hours on Tue Feb 26, 7-9pm, Winthrop D-Hall.
• (2/20) The handout about compactness (with completed proofs) is available.
• (2/14) Prof. Auroux's office hours on Tuesday 2/19 are cancelled. Class time will be devoted to working with the CAs to prove the equivalence between different notions of compactness. (A worksheet will be handed out).
• (2/14) Emily Saunders is holding an extra section tonight in SC 222 from 7:30-8:45pm. (One time only).
• (2/9) Valerie Zhang's section is moving to Tuesdays just after lecture (1:30-2:45pm in SC 222).
• (2/3) The office hours and section times of the course assistants are now set (see above). Also, Emily Saunders will be running a proof writing workshop on Tuesday 2/5 in SC 411 for those of you who are new to writing proofs.
• (1/15) This website is live, and the syllabus is posted.

Homework

Homework assignments will be posted here. You are encouraged to discuss the homework problems with other students. However, the homework that you hand in should reflect your own understanding of the material. You are NOT allowed to copy solutions from other students or other sources.

No late homeworks will be accepted. However, we will drop your lowest homework score, so you are allowed to miss one assignment without a penalty.

All homework submissions should be uploaded to Canvas (handwritten work is welcome, but please upload a scan or photo).

• Homework 1 due Thursday Feb 7 (.tex) and solutions (.tex)
• Homework 2 due Thursday Feb 14 (.tex) and solutions (.tex)
• Homework 3 due Tuesday Feb 26 (.tex) and solutions (.tex)
• Homework 4 due Tuesday March 5 (.tex) and solutions (.tex)
• Homework 5 due Thursday March 14 (.tex) and solutions (.tex)
• Homework 6 due Tuesday April 2 (.tex) and solutions (.tex)
• Homework 7 due Tuesday April 9 (.tex) and solutions (.tex)
• Homework 8 due Thurday April 18 (.tex) and solutions (.tex)
• Homework 9 due Tuesday April 30 (.tex) and solutions (.tex)

Exams

Final:

The final exam will take place on Monday May 13, 9:00-12:00, in Emerson 210. It will cover all of the material seen during the semester. Rudin allowed, no other materials allowed.

See here for more information about the content covered in the exam and additional practice problems from Rudin.

Midterm:

The midterm took place on Tuesday March 12, 12:00-1:15, in Science Center Hall E (usual place and time). It covered the material seen in lecture up to Tuesday March 5 (most of it included) -- specifically, Rudin pages 1-63, minus the appendix to Chapter 1 and the section on perfect sets on p.41-42.

Allowed: Rudin's book but NO OTHER MATERIALS (no notes, no calculators, no electronics). IMPORTANT: to take advantage of this policy, you need a physical copy of the book that isn't heavily annotated with extra text! (highlighting/underlining is fine). (or a printout / photocopy of chapters 1-3).

• Midterm solutions
• Midterm practice problems
• Solutions to the practice problems

Midterm score distribution: the median score was 90 out of 120, the lower quartile was 74, the upper quartile was 104. This means: 25% of the scores were below 74, 25% between 74 and 90, 25% between 90 and 104, 25% above 104. Scores are available on Canvas; you can get your midterm back in class on Thursday 3/28, or during office hours.

Syllabus

Note: the page numbers shown for each day's material are approximate. The actual contents covered in each lecture may end up being slightly ahead or slightly behind of schedule.

Date	Topics	Book / HW
Tue 1/29	Ordered sets, least-upper-bound property; fields.	p.1-8
Thu 1/31	Real numbers; complex numbers; Euclidean spaces.	p.9-18
Tue 2/5	Functions; finite, countable, uncountable sets; unions and intersections.	p.24-30
Thu 2/7	Metric spaces; neighborhoods, open sets; limit points, closed sets.	p.30-34 / HW1 due
Tue 2/12	Limit points and closed sets; interior and closure.	p.34-36
Thu 2/14	Compact sets.	p.36-38 / HW2 due
Tue 2/19	Review + work session: equivalent notions of compactness.	(handout)
Thu 2/21	Compact subsets of Rk; connected sets.	p.38-40, 42-43
Tue 2/26	Convergent sequences; Cauchy sequences.	p.47-54 / HW3 due
Thu 2/28	Subsequences; monotonic sequences in R; upper and lower limits; special sequences.	p.51-58
Tue 3/5	Series, comparison criterion; root and ratio tests.	p.59-68 / HW4 due
Thu 3/7	Power series; absolute convergence; summation by parts; rearrangements.	p.69-78
Tue 3/12	MIDTERM.
Thu 3/14	Limits of functions; continuity.	p.83-88 / HW5 due
	(SPRING BREAK)
Tue 3/26	Continuity and compactness; continuity and connectedness.	p.89-93
Thu 3/28	Left and right limits; discontinuities of monotonic functions.	p.94-98
Tue 4/2	Differentiable functions; mean value theorems; L'Hôpital's rule.	p.103-110 / HW6 due
Thu 4/4	Taylor's theorem; Riemann integrals; integrability.	p.110-113, 120-123
Tue 4/9	Integrability criteria; properties of the integral.	p.123-131 / HW7 due
Thu 4/11	Change of variables; integration and differentiation.	p.131-134
Tue 4/16	Sequences and series of functions. Uniform convergence. Uniform convergence and continuity.	p.143-151
Thu 4/18	Uniform convergence vs. integration and differentiation.	p.150-154 / HW8 due
Tue 4/23	Equicontinuity. Stone-Weierstrass theorem.	p.155-164
Thu 4/25	Power series, exponential, logarithm, trigonometric functions.	p.172-174, 178-184
Tue 4/30	Fourier series.	p.185-191 / HW9 due
Mon 5/13	FINAL EXAM (9:00-12:00 in Emerson 210)