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I'm starting to read Baby Rudin (Principles of mathematical analysis) now and I wonder whether you know of any companions to it. Another supplementary book would do too. I tried Silvia's notes, but I found them a bit too "logical" so to say. Are they good? What else do you recommend?

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A couple of things here to help you get better answers to a fair question. First, edit this post to make it community wiki. While you're at it, edit the question to include things such as some description of your level of familiarity with theorem-proof style mathematics, and where you are intending to end up in the short, medium, and longer term. Also, are you in an advanced calc class right now? – Tom Stephens Aug 19 '10 at 9:40
You may want to look at on George M. Bergman's site. – Pierre-Yves Gaillard Aug 19 '10 at 10:09

6 Answers 6

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.

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+1. This is an absolute must for anyone studying real analysis seriously and now that it's in Dover,there's no good reason not to have a copy. – Mathemagician1234 May 5 '12 at 17:47

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!!

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

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+1 for 2 excellent recommendations. Korner in particular works very well filling out the terseness of Rudin. – Mathemagician1234 Apr 8 '13 at 17:43

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

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There is a set of notes and additional exercises due to George Bergman. See his web page...

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Dear GEdgar: Please look at my comment above. – Pierre-Yves Gaillard Aug 19 '10 at 11:09
Nice link. I like it – Digital Gal Aug 19 '10 at 13:33
I'm not familiar with these. But I found Bergman's companion to Lang's Algebra quite useful when I was studying from that book, so consider that a qualified recommendation for Bergman's course materials in general. – Michael Lugo Aug 19 '10 at 16:50

Okay, so as you haven't accepted any of the answer, I thought maybe I should give a try to answer as best as I can.

I'm currently doing Chapter-3 from Walter Rudin's text, and simultaneously following these wonderful notes, by UC,Davis.
They include proof of each and every theorem and lemma in the book, and even a statement that is left in the book for the readers to prove is proved in these notes.

If you want, you can follow only these notes and not do theory from the book at all, and directly come back to it to solve problems at the end.

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