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:
- Just follow Abbott’s book as is, and forget about using metric spaces.
- 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).
- Supplement Abbott with a cheap textbook (roughly $10) on analysis like Rosenlicht.
- Supplement Abbott with something like Kaplansky’s text on metric spaces ($30).
- 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.”