[Book] Mathematical Proof

Can someone with a strong background in mathematics recommend a good book examining formal proofs with breakdown and explanation? Something like/the equivalent of "3000 Solved Problems in Calculus"?

One of the young programmers I work with wants to learn the "grit" behind formal proofs. I'm not nearly as experienced with mathematics as I am with programming. My explanations are all visual demonstrations. The demonstrations are apparently not very satisfying.

*shrug*

I usually just assume the mathematicians know what they are talking about.

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Sorry. Feel free to move this around...

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Soma