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? 


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.
