Here you can find all 26 lectures of my Real Analysis course at Harvey Mudd College. These lectures were taped in Spring 2010 with the help of Ryan Muller and Neal Pisenti.

### Lectures

The entire course is assembled as a playlist on YouTube. And below are links to individual lectures.

Lecture 1: Constructing the rational numbers

Lecture 2: Properties of Q

Lecture 3: Construction of R

Lecture 4: The Least Upper Bound Property

Lecture 5: Complex Numbers

Lecture 6: The Principle of Induction

Lecture 7: Countable and Uncountable Sets

Lecture 8: Cantor Diagonalization, Metric Spaces

Lecture 9: Limit Points

Lecture 10: Relationship b/t open and closed sets

Lecture 11: Compact Sets

Lecture 12: Relationship b/t compact, closed sets

Lecture 13: Compactness, Heine-Borel Theorem

Lecture 14: Connected Sets, Cantor Sets

Lecture 15: Convergence of Sequences

Lecture 16: Subsequences, Cauchy Sequences

Lecture 17: Complete Spaces

Lecture 18: Series

Lecture 19: Series Convergence Tests

Lecture 20: Functions – Limits and Continuity

Lecture 21: Continuous Functions

Lecture 22: Uniform Continuity

Lecture 23: Discontinuous Functions

Lecture 24: The Derivative, Mean Value Theorem

Lecture 25: Taylor’s Theorem

Lecture 26: Ordinal Numbers, Transfinite Induction

The text for the course was *Principles of Mathematical Analysis* by Walter Rudin, but you do not need the text to follow these lectures. Also, I realize the board is hard to read, so I’ve supplied some linked lecture notes in the tab above (they may not align perfectly with these lectures, since it’s from a different semester. You can also see my course webpage from the last time I taught the course, with homework schedule.)

### FAQ

#### Why did you make these videos?

My student Ryan Muller was developing an online learning platform and asked if I’d be interested in recording my Analysis lectures as YouTube videos to embed in it. After some initial hesitancy, I agreed. But after putting them up, I began getting appreciative e-mails from around the world. So I decided to leave them on YouTube.

#### What textbook do you use?

We use the classic text *Principles of Mathematical Analysis* by Walter Rudin, and cover the first half of the textbook (Chapter 1 to 5, and part of Chapter 7). You can see what problems I assigned from the last time I taught the course, at my course webpage, linked above.

#### I wish the resolution were better.

Me too. But that was the technology we had… the recording equipment (mounted in the ceiling of the lecture hall) only records at a lower resolution. But check out my course notes, linked above, if you would like to see some notes that loosely track with the videos.

#### Is your course a MOOC?

Technically, no. We made these videos before the word MOOC (massively open online classes) became a buzzword. Usually a MOOC has a participation and feedback component and involves a large number of students all working on the same course schedule. My real analysis videos are intended to supplement your own learning.

#### Have you been using the videos in a ‘flipped classroom’?

No, but you are welcome to. One downside to the videos is that the resolution is not that great, so the blackboard writing is hard to read. Also, the material has not been broken into the bite-size chunks that some flipped classroom models use. But I imagine some instructors have been using it this way, nonetheless — it has been neat to see how popular the videos have been in Sierra Leone. 🙂

#### Why do you discuss mathematical writing in this course?

Learning to communicate mathematics effectively is one of the goals of my Real Analysis course at Harvey Mudd College. This handout “Guidelines for Good Mathematical Writing“ is my guide to help students think about their writing.

#### Are you going to do videos for Analysis II?

There are no current plans to do this. We would first need better resources for classroom video capture. But I may do other online projects… stay tuned. 🙂

#### How do I stay tuned to your other projects?

You can follow me on Twitter. Or check my homepage periodically. Or my author webpage.

#### Can I write you e-mail?

I welcome e-mail, especially stories of how you found the videos useful in your own life situation. If you write, please tell me the country you are writing from, and a little bit about your situation and background. (Are you in college, or have no access to college? Are you a non-traditional student? etc.) Although I may not be able to respond, I do read every e-mail carefully.

#### How can I thank you?

If you found the videos helpful, you can ‘like’ the videos on YouTube. And for a good read about mathematics, you might enjoy my book.