Companions to Rudin?

Solution 1:

1) Introduction to real analysis by Bartle and Sherbert

2) Methods of Real Analysis by R.R. Goldberg

3) Mathematical Analysis by Tom Apostol

4) Real and Abstract Analysis by Karl Stromberg.

5) A radical approach to real analysis by David M Bressoud by MAA.

The first book is a very good book for a beginner. The next two are classics. (4) is also very good in case you want to read something advanced. The last one keeps entertaining you with some interesting examples as well as some interesting history of Real Analysis.

Happy Reading!!

Solution 2:

Gelbaum and Olmsted, Counterexamples in Analysis.

The first real analysis/advanced calculus class is full of theorems with multiple conditions, and it can be difficult to tell which ones are necessary for what parts of the theorem. This book will provide examples for why the theorems are as they are and not otherwise.

Solution 3:

The book Understanding Analysis by Stephen Abbott is very good. So is A Companion to Analysis by T. W. Körner.

Solution 4:

I find Terrence Tao's notes to be a great companion, but he deviates from the order Rudin's presented in. Course 1 course 2