What books can help me master the exercises in Rudin's Principles of Mathematical Analysis? I'll  major in math at a local University. I've finished High School Calculus and I've self-studied very basic Multivariable Calculus, Linear Algebra, and Real Analysis. I've watched video lectures of them.
While I can read and follow most of the proofs and chapters in Rudin's PMA, I cannot do the exercises. I thought that PMA was an introductory book to Real Analysis so I thought I shouldn't have too much trouble. But I am having a lot, perhaps because I was not exposed to rigorous proofs (I've seen a lot but I don't know how to formulate one myself). I searched and saw many recommendations for How to Prove It by Velleman and another book by Polya. 


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*Will reading one of it be fine as a bridge from non-rigorous high school math to rigorous University-level math like PMA? 

*Will  these books help me with Rudin's PMA's exercises? 

*Are there any books that help as bridges which are not the "proving-type"? 

*Finally, when does one typically take a Real Analysis course with Rudin's PMA in University? 

 A: As others have pointed out, Rudin's book is a little bit hard even for people who has more mathematical maturity. You should try some other alternatives, for example:


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*Bryant's: And Yet Another Introduction To Analysis
It's perhaps a good idea to use:


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*Hairer/Wanner's: Analysis by It's History
This book explained me a lot about the hierarchy of the proofs in Analysis and what were the challenges met by the people who created it. Another interesting read is:


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*Bressound's: A Radical Approach to Real Analysis.


Also, take a look at some of the recommendations in the MAA Reviews. One interesting review is the one on Rudin's book. I'd follow Arnold's advice, the book he recommends is superb. 
Now beyond these historical perspectives on analysis, you might find this book useful:


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*Abbott's:Understanding Anaysis
You might also like the following book:


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*Stillwell's: The Real Numbers
It's not too related to analysis in the sense of Rudin's book, but I think it's illuminating for the subject.
A: 
I've self-studied very basic Multivariable Calculus, Linear Algebra, and Real Analysis. I've watched video lectures of them.

and

While I can read and follow most of the proofs and chapters in Rudin's PMA, I cannot do the exercises. 

Most Multivariable Calculus and Linear Algebra books provide exercises for the readers to do themselves.  Have you done those exercises while you self-studied them ? Or you just skip them?
If you had skipped them, that's probably why you had trouble doing the exercises in PMA because you have not done enough exercises. If you can read and follow most of the proofs and chapters in Rudin's PMA, I believe you have been exposed enough rigorous proofs.  It's time to do them yourself.
Take the Multivariable Calculus textbook you were using, do the exercises. If you have trouble doing them, ask questions here. There are plenty of experts here willing to help you.
Once you have done enough exercises in Multivariable Calculus, then you come back to PMA and do the proofs of those exercises. You will feel much easier then.
