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I am taking real analysis in university. I find that it is difficult to prove some certain questions. What I want to ask is:

  • How do we come out with a proof? Do we use some intuitive idea first and then write it down formally?
  • What books do you recommended for an undergraduate who is studying real analysis? Are there any books which explain the motivation of theorems?
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The best book I know of for "explaining the motivation of theorems" in elementary real analysis is Victor Bryant's 1990 book Yet Another Introduction to Analysis. Moreover, I think this book is strongest when used as supplementary reading for students, which is your situation. – Dave L. Renfro Mar 1 '13 at 15:45
up vote 11 down vote accepted

I have been teaching about 13 years in collage so I have seen many books or texts written by for example, Rudin, Bartle, Apostol and Aliprantis in Analysis. But the ones have been useful for me or for students that have topological approaches or graphical approaches. Rudin's is a great one but there is not examples as you find variously in Apostol's. Bartle's some chapters are including figures but two last ones have a few. Aliprantis's is full of problems and because of that I prefer it. Just an advice : If you are new in any field and that's why you want to be more familiar to those new concepts; try to select the books whose have many solved problems. I prefer to teach through practice. Sorry if my written in English is not good as others.

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You're English is just fine! + – amWhy Mar 1 '13 at 20:33

While this doesn't speak, directly, to Real Analysis, it is a recommendation that will help you there, and in other courses you're encounter, or will encounter soon:

In terms of both reading and writing proofs, in general, an excellent book to work through and/or have as a reference is Velleman's great text How to Prove It: A Structured Approach. The best way to overcome doubt and apprehension about proofs, whether trying to understand them or to write them, is to be patient, persist, and dig in and do it! (often, write and rewrite, read and reread, until you're convinced and you're convinced you can convince others!)

One helpful (and free-to-use) online resource is the website maintained by Interactive Real Analysis.

"Interactive Real Analysis is an online, interactive textbook for Real Analysis or Advanced Calculus in one real variable. It deals with sets, sequences, series, continuity, differentiability, integrability (Riemann and Lebesgue), topology, power series, and more."

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I wish I could have this new one.+ – Babak S. Mar 1 '13 at 20:58
It's a very good book, @Babak . I should have also posted a link to Mason, Burton, and Stacey's Thinking Mathematically, which I just did here in the comment here. It's good supplementary reading, but "How to Prove it" is most relevant; the added link would be good to do on the "side". – amWhy Mar 1 '13 at 21:10

To come out with a proof I pretty much always started by 1. imagining a specific example 2. drawing the example as picture if possible 3. persuading myself (by looking at the picture) that the thing we were being asked to prove was actually true, then 4. making up some notation to describe what I was looking at.

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I started studying Rudin's Principles of Mathematical Analysis book for analysis. I just couldn't understand it because I was thinking that definitions are some decorative blah blahs and proofs are some unintelligible mathematician sorceries.

Then I have decided to learn something about proofs from the book An Introduction to Mathematical Reasoning. Just after three chapters all begin to make sense. It is a perfect book for people like me.

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