**Spring 2020**

**Office Hours:** WED 8:30 – 9:30am and WED 2:30–3:30pm, or by appointment.

**Graders: **Hanna Hoffman, Deyana Marsh**Tutoring Hours: **TUE 8-9:30pm

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.**

See also this guide. Rewrites should be handed in in parts.

### 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.

## Midterms and Grading

There will be three exams:

- Midterm 1: Take-home available Feb 24 in class, due Fri Feb 28 at 5pm.
~~Midterm 2: Take-home available Mar 30 in class, due Apr 3~~[COVID CHANGE]- Final: Take-home due during the regularly scheduled final period.

Each of these and your homework average will count 25% of your course grade, with [COVID CHANGE] the highest component worth an additional 25%. The lowest two homework grades 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!

## 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! 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.*