## Math 331: Foundations of Analysis

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

Fall 2020.

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

Syllabus:  http://math.hws.edu/eck/courses/math331_f20.html

Monday, Wednesday, Friday, 1:20–2:20 PM
Room Eaton 111.

```

 Homework Homework 1,Due September 2, Homework 2,Due September 11, Homework 3,Due September 18, Homework 4,Due September 26,

### Fifth Week: September 21, 23, and 25

We are running a little behind the original schedule from the syllabus. We still need to finish the Extreme Value Theorem and uniform continuity, from Section 2.6, which were scheduled for last week. That will finish Chapter 2. After that, we will be taking a short excursion from the textbook before moving on to Chapter 3, by looking at metric spaces. We should be able to start metric spaces before the end of class on Wednesday. You should read the short introduction and the section on open and closed sets. The reading guide for the metric space material is included in

### Fourth Week: September 14, 16, and 18

We have not yet finished with limits. We still have some things from Section 2.3 to do, and all of Section 2.4 (although we will not spend much time on 2.4). But we should be able to turn to our next topic, continuity, on Wednesday. The new reading is Chapter 2, Sections 5 and 6. Those sections cover continuity and prove two important theorems about continuity on a closed interval: the Intermediate Value Theorem and the Extreme Value Theorem (called the Max-Min Theorem in the textbook). Also covered is a stronger form of continuity on an interval called uniform continuity, and it is proved that any continuous function on a closed, bounded interval is in fact uniformly continuous. The reading guide for the material on continuity is

### Third Week: September 7, 9 and 11

After finishing up the Heine-Borel and Bolzano-Weirstrass theorems, we will move on to Chapter 2, which covers limits and continuity. The reading for the week is Sections 2.0 through 2.4, but we will not spend any class time on 2.0 or 2.1, and we might not get to 2.4 until next week. For much of the rest of the semester, we will be looking at topics that you have already seen in Calculus I and II, but we will be taking a theoretical, proof-oriented approach that uses the rigorous view of the real number system that we have been pursuing. Section 2.2 and 2.3 cover the formal definition and theory of the limit of a function at a point, with proofs that use the epsilon-delta definition of limit. The reading guides for this material, including guide 6 continued from last week, are

Homework 2 is due on Friday. Homework should be turned in on time. However, if circumstances make that difficult for you, you should consult with me about getting an extension.

### Second Week: August 31; September 2 and 4

We still have to finish up some material from Section 1.2 on the Least Upper Bound Property and the Archimedian Property of the real numbers. The new reading for the week is Sections 1.3 and 1.4. Sections 1.3 gives a set of axioms for the real numbers as a "complete, ordered field." Section 1.4 covers two important theorems about the real numbers: the Heine-Borel theorem and the Bolzano-Weirstrass theorem. That will probably take us into the beginning of next week. The "reading guides" for this material are

My office hours on Zoom this week will be Tuesday and Thursday, 12:30 to 2:00 and 6:30 to 8:00. You can use the Canvas Calendar feature to make an appointment during those times. Note that I will usually be in my office between classes on Monday, Wednesday, and Friday, from about 11:15 to 1:00. If you want to meet either on Zoom or in person during those times, you should arrange an appointment by email or by talking to me after class, or you should call my office (315-781-3398) to see whether I am available.

### First Week: August 24, 26, and 28

Welcome to the course!

The reading for the week is Chapter 1, Sections 1.0 through 1.2. Please read them! We might start covering Section 1.3 in class on Friday. From Section 1.1, you should be familiar with irrational numbers and the Fundamental Theorem of Arithmetic, but we will not spend a great deal of time on that. Section 1.2 gives a "construction" of the set of real numbers; that is, it defines a specific mathematical object that has the properties that we expect the real numbers to have.

I plan to write short guides to the readings for each lecture, emphasizing the most important parts in the reading and sometimes adding information or perspective that is not in the textbook. The guides for the first week are here:

The first homework assignment is already available. You are encouraged to write up your solutions in LaTeX, using a free account at overleaf.com, but I will also accept handwritten work scanned to a PDF file, at least at first. For more information about submitting homework, see

http://math.hws.edu/eck/math331/f20/submitting-homework.html