## Teaching Real Analysis

I am teaching real analysis for the second time in the fall, and I am excited about it. I used Stephen Abbott’s Understanding Analysis when I taught it in Fall 2011, and the students and I both loved it.

My one issue with it, though, is that I would rather do more with metric spaces (Abbott works with sequences in the real numbers as a foundation for the course); I found that I would often draw pictures of R^2 on the board to illustrate ideas relating to distance, and I would like to leverage this slightly more.

I am sold on the idea of using Abbott: it works ridiculously well for my flipped classroom, the students love it, and I am already familiar with it (I am hoping to stop completely redesigning every course I teach from scratch). Here are my ideas for incorporating metric spaces more:

1. Just follow Abbott’s book as is, and forget about using metric spaces.
2. Start the semester by looking at Abbott’s brief chapter on metric spaces (in Chapter 8), let students know that we are mainly going to be using it for examples in class, and they are not very responsible for knowing it (perhaps I might give challenge problems where they generalize results in terms of metric spaces, but not every student would need to do that).
3. Supplement Abbott with a cheap textbook (roughly \$10) on analysis like Rosenlicht.
4. Supplement Abbott with something like Kaplansky’s text on metric spaces (\$30).
5. Supplement Abbott with something like Keith Conrad’s notes on metric spaces (free).

Money is a factor, so I don’t want an expensive supplement.

I am mainly looking for comments like “It is a bad idea to try to integrate metric spaces with Abbott” or “It is a good idea, and here is the perfect source.”

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### 7 Responses to “Teaching Real Analysis”

1. Whitney L. Says:

Bret, Although I loved Real Analysis with you, I found that it was too easy. I had a lot of catch up to do for my graduate class. I may be unique in this situation. Maybe pull aside the stronger students in the class to do metric spaces. With the flip maybe it would be possible to have some students do this?

• bretbenesh Says:

Yep—it is not a great class to prep for graduate school. One thought that I have had is that if I did at least some integration of metric spaces into real analysis, I could offer problems at different levels. That is, for each topic (e.g. Continuous Functions), I could give something like four levels of problems for them to work through:

D-level: Students create an example (and probably a nonexample, too) of a continuous function with properties X, Y, and Z.
C-level: Students prove some result about a very particular function.
B-level: Students prove a general result about a topic.
A-level: Students general the topic to metric spaces.

This is obviously _very_ rough, but having the students progress through each topic would make it so that the average student does not need to know much about very general results, but students destined for graduate school would get a little closer to where they need to be.

Thoughts?

2. Darci Kracht Says:

I have never taught real analysis, so I have nothing to offer in the way of an answer to your question. However, I just learned about some software that you might find useful. It is called Perusall and was developed by Harvard physicist Eric Mazur. You load a pdf of your text or other materials. Students annotate it with questions or comments and respond to other students’ questions. The software assigns students a grade based on the quality of their annotations. It also gives the instructor a short list of the most frequently-misunderstood ideas, so you can address them in class. https://perusall.com/

• bretbenesh Says:

Mazur is awesome! I hadn’t heard of Perusall, but I am looking forward to playing with it more. It does not look like it includes Springer, which is Abbott’s publisher, but maybe that isn’t actually a problem.

Thanks!

3. Alok singh Says:

Pugh’s chapter 2 is all about metric spaces and the best intro I’ve ever seen for them.

• bretbenesh Says:

Thanks! I have looked at Pugh, and ultimately thought that it probably isn’t the right choice for my class (Honors Analysis at Berkeley is higher than the level at my school).

Do you think it is worth having them spending twice as much on textbooks for the class? Pugh costs about the same as Abbott. Or maybe I could copy some of it as Fair Use?

4. oldmannick Says:

I imagine you could get it as fair use. I really doubt that Pugh will care (he was my real analysis professor, and I didn’t take the honors version).

I think it’s worth understanding metric spaces because they simplify a lot about basic analysis by showing that limits just ask that you get close to a point, and what “close” means can be generalized to interesting spaces like graphs and trees for the computer scientist types among them.

I think they’ll end up looking at libgen.io for their books if they have any sense.