# Math 331: Foundations of Analysis

```   Department of Mathematics and Computer Science
Hobart and William Smith Colleges

Fall 2019.

Instructor:  David J. Eck  (eck@hws.edu)

Monday, Wednesday, Friday, 1:30–2:30 PM
Room Gulick 223.

```

### About The Course and the Textbook

Math 331 is a first course in real analysis. For much of the course, we will be talking about topics that you covered in Calculus I and Calculus II: limits, continuity, differentiation, integration, and infinite series and sequences. It is, however, decidedly not just another calculus course. In your calculus courses, you mostly learned some techniques and used them to solve some problems. For example, you memorized a list of rules for differentiation, and you used them to find the derivatives of a lot of functions (often given by formulas that are unlikely ever to turn up in a real-world application). You might have done some simple proofs, and you probably spent some time developing intuition about why certain things are true, but for the most part, "calculus" meant "calculating" based on some memorized formulas.

In this course, you will do very little calculating. Instead, you will be doing mathematics. That is, we will start from first principles and build up a theory from there, based on rigorous logical reasoning. The first principles are the definition and basic properties of the real numbers. The theory that we will develop is the real Calculus, the theory of the infinite and the infinitesimal that has taken some 2500 years to develop.

The textbook for the course is Foundations of Analysis by David Belding and Kevin Mitchell. The two authors were professors at Hobart and William Smith Colleges until their retirement in 2017. This book was written specifically as a textbook for a pair of courses that they developed, Foundations of Analysis I and II. We will cover chapters 1 through 4 of the book.

The textbook was written before Math 135 was invented, and the courses based on it were often the first experience that students had with rigorous mathematics. You should be more prepared for the rigor than those earlier students were. I plan to take advantage of that fact to move beyond the textbook and cover more general ideas in a few cases. In particular, we will look at metric spaces, which generalize the idea of distance between real numbers. The reading for this material will be in the form of handouts. Most of the extra material will be in the first half of the course.

### Assignments and Presentation

Homework assignments will be collected weekly. Most of the homework exercises will be proofs, ranging from fairly simple to more complex. I will often give you a chance to rewrite proofs, but only when the first attempt is serious.

You are allowed and encouraged to discuss homework assignments with other people in the course. However, you should write your own solutions in your own words.

I will sometimes ask people to present proofs in class, either proofs from the homework that everyone has worked on or proofs that are assigned individually. Everyone in the class should expect to do a presentation at least once during the semester. In addition to presentations in class, I might sometimes ask you to come to my office hours to discuss or present the proofs that you have submitted for homework.

### Tests and Final Exam

There will be two in-class tests that will cover mostly definitions and basic concepts, including some simple proofs. The two in-class tests will be on Wednesday, September 25 and Friday, November 8.

There will also be a take-home midterm exam, which will be distributed in class soon after Fall break.

The scheduled final exam period for this course is Sunday, December 15, from 8:30 to 11:30. There will be a final exam, which will likely consists of both an in-class part and a take-home part. (We might, however, decide to eliminate one of those parts.)

```             First in-class test:      15%
Second in-class test:     15%
Take-home midterm:        15%
Final exam:               20%
Homework:                 35%
```

Letter grades are assigned approximately as follows: 90-100: A; 80-89: B; 65-79: C; 55-64: D; 0-54: F. Grades near a cutoff get a + or -. (Note that I sometimes curve a numeric grade upwards when it seems that the grades do not reflect the level of ability of the class.)

### Attendance Policy

I am supposed to announce my attendance policy. Attendance—and arriving to class on time—is required, except in extraordinary circumstances. But the attendance policy is not enforced except by admonishment, dirty looks, and possibly public shaming.

### Statements from the Center for Teaching and Learning

Disability Accommodations: If you are a student with a disability for which you may need accommodations, you should self-identify, provide appropriate documentation of your disability, and register for services with Disability Services at the Center for Teaching and Learning (CTL). Disability related accommodations and services generally will not be provided until the registration and documentation process is complete. The guidelines for documenting disabilities can be found at the following website: http://www.hws.edu/academics/ctl/disability_services.aspx. Please direct questions about this process or Disability Services at HWS to Christen Davis, Coordinator of Disability Services, at ctl@hws.edu or x 3351.

### Office Hours, E-mail, WWW

My office is room 313 in Lansing Hall. My office phone extension is 3398. I am on campus most days, and you are welcome to come in anytime you can find me there. I will announce regular office hours and post them on my office door and on the course web page as soon as my schedule is determined, but note that your office visits are certainly not restricted to my regular office hours!

My e-mail address is eck@hws.edu. The home page for this course on the web is located at http://math.hws.edu/eck/math331/index_f19.html. This page will contain a weekly guide to the course, including homework assignments. You will want to bookmark this page! This course does not use Canvas.

### Tentative Schedule

Here is a very tentative schedule for the course. I will try to keep approximately to this schedule, but the actual schedule will be posted weekly on the course web page.

Aug. 26, 28, 30 The real numbers; Dedekind cuts (Sections 1.0–1.2)
Sept. 2, 4, 6 Axioms for the real numbers (Section 1.3)
Metric spaces; open and closed sets (Handout)
Sept. 9, 11, 13 Heine-Borel & Bolzano-Weirstrass theorems (Section 1.4)
Compactness in metric spaces (Handout)
Sept. 16, 18, 20 Sequences and limits of sequences (Sections 4.0 and 4.1)
Functions and limits of functions (Sections 2.0–2.2)
Sept. 23, 25, 27 Limit theory (Section 2.3)
In-class test on Wednesday, September 25
Sept. 30; Oct. 2, 4 Continuity (Section 2.5)
Sequences & Continuity in Metric Spaces (Handout)
Oct. 7, 9, 11 Continuity on closed intervals (Section 2.6)
Connected Sets (Handout)
Oct. 16, 18 The derivative (Sections 3.0–3.2)
Take-home test will be handed out
No class on Monday because of Fall Break
Oct. 21, 23, 25 Mean Value Theorem (Section 3.3)
The Riemann Integral (Section 3.4)
Oct. 28, 30; Nov. 1 Properties of Integration (Section 3.5)
Fundamental Theorem of Calculus (Section 3.6)
Nov. 4, 6, 8 Taylor's theorem with remainder (Section 3.7)
In-class Test on Friday, November 8
Nov. 11, 13, 15 Monotone and Cauchy Sequences (Section 4.2)
Complete Metric Spaces (Handout)
Nov. 18, 20, 22 Infinite Series & Convergence Tests (Sections 4.3, 4.4)
Thanksgiving Break, November 25–29
Dec. 2, 4, 6 Sequences and series of functions (Sections 4.5, 4.6)
December 9 Last day of class; wrap up the course.
December 15 Final Exam, Sunday December 15, 8:30 AM