Originally Posted by

**win**
In my opinion, given all the courses you've mentioned, you should be very well prepared. While there is always more you can learn, I would suggest focusing more on making sure to learn the topics in your classes deeply and thoroguhly as opposed to trying to cover more subjects. Even at the expense of other topics. For example, learning the material in these courses you initially listed should take priority over auditing an optimization course, which is a nice bonus but something that you can do with relative ease if you really understand the math.

As for books suggestions, Rudin's Principles is the obvious choice, but I think most people's first encounter is a little intimidating given its sparing prose, lack of examples, and so on. I think once you are in class you will find that less the case, but I think two useful bridge books are Understanding Analysis by Abbott, which is a gentler introduction which I think is more accessibly written. It wont replace rudin, but it would probably be an easier introduction to the most important topics.

Another one, which is not specifically analysis, but I think is very helpful is Thomas Sibley's Foundations of Mathematics. I have not come across an electronic version (if you do, let me know), but it is really a very accessible introduction to a lot of the topics that rudin either takes for granted or moves through quickly but are fundamental later on. For example, it spends a fair about of time explaining the basics of set theory, of how to write proofs, the foundations of functions and relations.

Anyways, it sounds like you are well covered. Good luck with your courses.