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There are many books, written by highly decorated academics, which feature proofs that I can hardly comprehend in an acceptable amount of time. Roughly each week, it happens that I find myself having spent around three hours to convince myself of a single proof - usually, the proofs are from textbooks for undergraduates. Sometimes, I am horribly slow at finding detailed arguments which connect the presented conclusions (and which are necessary to convince myself that the conclusions are beyond question). Then, I feel that the authors take steps that are not small enough for me; and I wonder whether I should digest every proof (or merely use the theorems).

I am not particularly gifted. Still, I strive to improve myself. Do you think that one can learn or train to understand proofs faster?

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Do a mixture of both strategies. If you try to prove every statement in mathematics, then you will move at a snails pace. For instance, can't you just accept Fermat's Last Theorem without fully understanding 200 pages of the most advanced math in the world? On the other hand if you prove nothing, then you are missing out on the fun and not learning very much. There's a trade off.

Also, say you've proven every theorem in a vector analysis book but have forgotten the proofs and the formulas. Are you going to reprove every statement you meet again and have forgotten about? Simply pick and choose which ones you will do that for.

How do you choose what to prove, what not to? You could start at a section in the book that interests you and prove each statement that section depends on.

In conclusion, there is simply too much mathematics in the world to prove every statement you come across, while on the other hand you need to learn the proof techniques and know more about certain proofs than others, so use your big, powerfully evolved brain to choose wisely.

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I think you can try the book "How to study as a Mathematics Major" by Lara Alcock, published by Oxford university press. She has some advices on how to understand proofs efficiently and possibly construct own proofs. The book is based on the results from mathematics education.

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