This course endedon December 6, 2022 |

## Math 331: Foundations of Analysis

Department of Mathematics and Computer Science Hobart and William Smith Colleges Fall 2022. Instructor: David J. Eck (eck@hws.edu) Syllabus: https://math.hws.edu/eck/courses/math331_f22.html Monday, Wednesday, Friday, 9:50–10:50. Room: Coxe 1. Regular Office Hours, Lansing 313: Monday, Wednesday, and Thursday 1:30 to 3:00. Usually Available: Tuesday, 1:00 to 3:30 and in the period after class.

Homework | |||

Homework 1,Due August 31. |
Homework 2,Due September 7. |
Homework 3,Due September 19. |
Homework 4,Due September 26. |

Takehome Test 1,Due October 3. |
Homework 5,Due October 7. |
Homework 6,Due October 17. |
Homework 7,Due October 24. |

Homework 8,Due November 2. |
Homework 9,Due November 11. |
Takehome Test 2,Due November 18. |
Homework 10.Be present for the last week of classes! |

### Final Presentations: December 8

The final exam period for this course is Wednesday, December 8, from 7:00 to 9:00 PM. We will use this period for final project presentations.

### Fifteenth Week: November 29 and 30; December 2

For the last week of the semester, we will turn again to metric spaces. After a short review of the material that we looked at previously, I will cover three new topics: Compactness, Completeness, and Connectedness.

### Fourteenth Week: November 21

Because of Thanksgiving break, we only have class on Monday this week. (Happy Thanksgiving!)

We will finish our look at series of functions, which will also finish Chapter 4. That's the last of what we will do from the textbook! We should also take some time to talk about final projects.

Resubmissions for Homework 9 are due on Wednesday, November 30.

### Thirteenth Week: November 14, 16, and 18

The second takehome exam is due at the start of class on Friday, November 18:

The second in-class test will be on Friday, November 18. Here is the study guide:

Second In-class Exam Study Guide

Aside from the tests, we will work on Section 4.6, Series of Functions. Here is a summary of the main results:

Note that all the things that were supposed to be due on November 11 have been postponed to Monday, November 14.

### Twelfth Week: November 7, 9, and 11

Here is web page that we looked at in class on Friday: Math 331, November 11

We have looked at convergence tests for series of real numbers from Sections 4.3 and 4.4. We will work on examples of applying those tests. The we move on to Section 4.5, which covers sequences of functions. For a summary of results from Section 4.5, see

And see Professor Mitchell's web site, mentioned at the end of that summary, for some interactive examples.

The second take-home test and second in-class are coming up. The take-home test will be distributed this Friday, and the in-class test is next Friday. With a lot coming up in the next two weeks, here is a schedule:

- Friday, November 11:
- Homework 9 and resubmission of homework 8 are due.
- Take-home test and study guide for in-class test to be distributed.
- Final project topic must be chosen (or one will be assigned).

- Friday, November 18:
- In-class test.
- Take-home test due at start of class.

### Eleventh Week: October 31; November 2 and 4

We covered monotone sequences and the Monotone Convergence Theorem from Section 4.2 last Friday. We will cover Cauchy sequences from that section on Monday, and we will then move on to infinite series, Sections 4.3 and possibly into Section 4.4 on Friday. Here is an outline of the material on sequences and series of real numbers:

Remember that Homework 8 and the revisions for Homework 7 are due on Wednesday.

### Tenth Week: October 24, 26, and 28

We will finish will finish Chapter 3 with a proof of the Second Fundamental Theorem of Calculus and a look at Taylor polynomials. And to finish integration, I want to briefly cover improper integrals, which are covered in the book only in Exercise 3.5.16. We will then move on to Chapter 4, which covers sequences and series. The reading for the week is sections 3.7, 4.0, and 4.1. It is possible that we will start Section 4.2 on Friday. Here is a summary of the material on Taylor Polynomials:

### Ninth Week: October 17, 19, and 21

We continue the study of the Riemann integral with Section 3.5 (properties of the integral) and Section 3.6 (Fundamental Theorems of Calculus). Here is a short summary of the results in those two sections:

Note that revisions for Homework 5 are due by Wednesday, October 19. Homework 6 is due this Monday, October 17, and Homework 7 is due next Monday, October 24.

### Eighth Week: October 12 and 14

There is no class on Monday because of Fall break.

This course requires a final project and presentation, which is due at the end of the semester. Everyone in the class needs a different topic. Here is more information about the project, including some ideas for possible topics:

We did not get to Section 3.3, which covers the Mean Value Theorem, so we will pick up there on Wednesday. However, I will not cover the generalized mean value theorem and L'Hôpital's rule from that section; that material will be left as reading, if you want to do it. We will then move on to Section 3.4, which defines the Riemann integral. Here is a summary of the definitions and basic theorems:

### Seventh Week: October 3, 5, and 7

There is an in-class test on Monday, and the take-home test is due at the same time. A short homework on metric spaces will be due on Friday, along with revisions for homework 4.

For Wednesday and Friday, we move on to Chapter 3. The reading for the week is Sections 3.1, 3.2, and 3.3. These sections cover the definition and basic properties of the derivative. The material will be familiar to you from Calculus I. Here is a short summary:

### Sixth Week: September 26, 28, and 30

It's testing time! A take-home test is available. An in-class test is scheduled for next Monday, October 3. The take-home will be due at the start of the in-class test on Monday, October 3.

There is also an in-class test on Monday, October 3. A study guide is available and will be handed out in class on Wednesday.

Homework 4 is due on Monday of this week (September 26), along with any revisions for Homework 3.

In lecture this week, we will continue with metric spaces. We still need to cover some material on Open and Closed Sets. After that we will move on to Limits and Continuity. This will include infinite sequences in metric spaces and their limits. For the case of sequences in the real numbers, you can also see Section 4.1 in the textbook. here is a short summary of the main facts:

### Fifth Week: September 19, 21, and 23

We mostly finished with Section 2.5, continuity, last week. This week we will cover continuity on closed intervals, Section 2.6, including two important theorems: the Intermediate Value Theorem and the Extreme Value Theorem. That will finish Chapter 2. After that, we will be taking a short break from the textbook to look at metric spaces. A metric space is a set, M, together with a function d(x,y) for x,y in M that generalizes the notion of distance in the real numbers. You should read the following on-line material about metric spaces: Introduction to metric spaces and Open and Closed Sets. Here are reading guides for continuity and for metric spaces:

In class on Friday, September 16, we looked at an example of an increasing function defined on the interval (0,1) that is continuous at every irrational number and has a jump discontinuity at every rational number. The web page

draws a very close approximation of the function. However, note that almost all of the jumps are too small to see, so that it really looks like the graph is constant on most subintervals. In fact, I can only see nine or ten jumps. The web page gives you some options to make a more satisfactory graph. Now, maybe you'd like to think about what the ranged of these functions look like.

### Fourth Week: September 12, 14, and 16

We will continue with limits, Sections 2.2 through 2.4. Section 2.4 briefly covers limits at infinity, infinite limits, and one-sided limits. The reading guide for Section 2.4 is here:

By the end of the week, we will at least start Section 2.5, on continuity. (I have dropped the idea of looking at Section 4.1 this week; we will get to infinite sequences after we finish Chapter 2 and cover some of the basics of metric spaces.)

The due date for Homework 3 has been extended to **Monday (September 19), at 3:00**. However,
revisions for Homework&nbs;2 are still due by 3:00 on Thursday (September 15).

### Third Week: September 5, 7, and 9

We begin the week with the proof of the Heine-Borel Theorem, the definition of accumulation point, and the Bolzano Weirstrass Theorem. That will complete Chapter 1. We will move on to Chapter 2, which covers differential calculus. The main reading is Section 2.1, on functions, and Section 2.2, on limits, although I will try to get started on Section 2.3 by the end of the week. Most of the material in Section 2.1 should be familiar to you, and we will only look at a few points from that section in class. Here is the reading guide for Sections 2.2 and 2.3:

### Second Week: August 29 and 31; September 2

The reading for the week is to finish Section 1.2, read Section 1.3, and start Section 1.4. After finishing Section 1.2, we will turn to an axiom system for the real numbers. I hope that by the end of the week, we will start Section 1.4, up to the proof of the Heine-Borel Theorem. The reading guides for this material can be found here (but I don't expect to cover the Bolzano-Weirstrass Theorem until next week):

### First Week: August 22, 24, and 26

Welcome to the course!

The reading for the week is Chapter 1, Sections 1.0 through 1.2. Please read them! We will continue with Section 1.2 next week. Note that you are not responsible for the parts of the reading that cover historical background. 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 theorem, and you are not responsible for the proof). 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 publish short guides to the readings, emphasizing the most important points and often adding information or perspective that is not in the textbook. Generally, there will only be one or two guides per week, but for this first week, there are three; the third reading guide will carry us into next week:

The reading guides are part of the required reading!

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 neatly handwritten work. For more information about submitting homework, see

http://math.hws.edu/eck/math331/f22/submitting-homework.html