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I am an engineer with tons of math courses behind me. However, all of them were applied wherein instead of giving us intuition, they gave us a toolbox. (If you encounter this then use that to solve.)

For my research (and as a requirement for grad school), I am required to be able to write and read proofs. These can range from elementary to SIAM journal level.

My research is fairly computational (Optimization and Linear Algebra) and I need proofs only to be able to understand which algorithms work, their convergence and to write proofs for my algorithms if I ever construct one.

I have 2 options now:

  1. Read a book like How to Prove it or How to read and do proofs.

  2. Read Baby Rudin and end it all forever. (Since, I believe that should be enough of proof doing for the rest of my career)

I don't see Baby Rudin's (with all its topology & calculus) helping me in my objectives. Am I missing something?

Which way should I go?

Owing to severe time constraints, I can do only one.

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    It sounds like working through chapters 1-3 of Velleman is necessary, but not sufficient, to develop the skills you want. If it goes quickly, good, you can move on. But what's good about Velleman is the deliberate and structured approach he takes.2012-03-10

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Doing proofs is a skill to be developed, not a set of knowledge to learn. You can't become good at doing proofs by reading a book. You need to practice proving things yourself.

Books like Rudin can be useful as a source of little problems to prove (eg, work some of the exercises and attempt to prove some of the theorems on your own before reading the proof in the book). However if you read passively you will not improve very much. Even if you know the all the ins and outs of every definition and theorem, and can recall every proof, you will still be less well-equiped to face new proofs than someone who reads less carefully but trys to prove things on their own.

I recommend "How to Solve It" by George Polya, "Solving Mathematical Problems: A Personal Perspective" by Terro Tao, and "The Cauchy Schwarz Master Class" by Michael Steele.

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    Hah, ok well it was worth a shot anyways. I still really recommend the 3 books I mention in my answer. Also for reference the book in the discrete math course I took was "Discrete Mathematics: Introduction to Mathematical Reasoning". I thought it was really good, but not everyone online agrees.2012-03-10