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I'm on math carreer and I'm finding it very difficult, I've read "How to do a demonstrarion" and "logic for math" books, but I think I could really use a book full of things like "How to prove that a set is dense in another set", "How to prove this recursive function is bounded", "How to use the Stone-Weierstrass theorem".

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In the first year of my studies I found

Daniel Solow:
How to read and do proofs an introduction to mathematical thought processes

Amazon Link

It really helped me understand the basics.

However, to learn the application of theorems, like Stone-Weierstrass, one has to see those in action. I don't think there is a book that incorporates strategies for every theorem one needs.

In any ordinary book on calculus there should be plenty of examples how to use those theorems. Reading those textbooks can improve our "intuition" on how to use theorems.

Finally remember this quote form Euclid:

There is no royal road to geometry.

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  • $\begingroup$ The quote is also attributed to Menaechmus. I imagine the "quotes" were invented later, since it can be bad for one's health to dis a king. $\endgroup$ Apr 20, 2016 at 7:40
  • $\begingroup$ Appreciated. Yes I've read that one, but once in my teachers book I saw something like: "Things to remember: to prove a set is closed, prove its complement is open" hah, that's what I want to hear, sometimes :/ $\endgroup$ Apr 20, 2016 at 7:55

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