Book Recommendations and Proofs for a First Course in Real Analysis I am taking real analysis in university. I find that it is difficult to prove some certain questions. What I want to ask is:


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*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? 


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 MathCS.org: 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."

