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Hmm, first off, if you're going to be a doctor, then you really don't need to read Spivak or proof books. If you're interesting in mathematics, then great, you should read the books But if you just want to know the material to pass the test, then Spivak really isn't necessary.
Not that I don't want you to read Spivak or Velleman, but I don't want to give you false information!
Anyway, I have read Polya's book and the book is not a proof book. It doesn't explain proof methods like induction and stuff. The book is actually written for future teachers to give them methods how to make coherent explanations and how to teach students the art of problem solving.
It's a good read, but it's not the book you're looking for!
Not that I don't want you to read Spivak or Velleman, but I don't want to give you false information!
Anyway, I have read Polya's book and the book is not a proof book. It doesn't explain proof methods like induction and stuff. The book is actually written for future teachers to give them methods how to make coherent explanations and how to teach students the art of problem solving.
It's a good read, but it's not the book you're looking for!