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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.

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

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

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

  4. Finally, when does one typically take a Real Analysis course with Rudin's PMA in University?
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    $\begingroup$ There's no clear-cut way to learn how to do analysis proofs. 'Introduction to Analysis' by William R. Wade was used at my university for the introductory analysis courses. 'Principles of Mathematical Analysis' by Walter Rudin seems to cover material that my university covered in the real analysis course and fourier analysis/lebesgue integration course. Personally, I bridged the non-rigorous high school math to pure math by taking university courses at a level I could follow and moving up from there. $\endgroup$
    – Santeri
    Aug 8, 2015 at 23:30
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    $\begingroup$ How to write proofs may be the main thing you need to study. ${}\qquad{}$ $\endgroup$ Aug 8, 2015 at 23:33
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    $\begingroup$ When one is expected to be able to handle PMA varies from program to program, and very much from country to country. In some cases never. Often, second to third year. Worry not, you are probably well ahead. $\endgroup$ Aug 9, 2015 at 0:25
  • $\begingroup$ Baby Rudin is a challenging book, especially for someone starting out. You could try Spivak's Calculus as an easier introduction to analysis. $\endgroup$
    – littleO
    Aug 9, 2015 at 0:40
  • $\begingroup$ A very basic bridge book is "Elementary Analysis: The Theory of Calculus" by Kenneth Ross. Spivak is also good. Baby Rudin is challenging even for university students. $\endgroup$
    – jeo15
    Aug 9, 2015 at 0:43

2 Answers 2

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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:

It's perhaps a good idea to use:

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:

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:

You might also like the following book:

It's not too related to analysis in the sense of Rudin's book, but I think it's illuminating for the subject.

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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.

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