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I have an exam tomorrow and I need to learn 23 theorems and their proof about measure theory and Fourier series, which I need to be able to reproduce. Can someone share some tips on how to learn proofs of theorems fast?

In previous years I would just understand a proof then write it out many times until I memorized it. But it used to be just 5-10.

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I can't really give you any tips except, and I don't mean this to be condescending, start studying earlier than the day before the exam. –  Owen Sizemore Aug 5 '13 at 16:51
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As I was studying, I learned the ideas of proofs. It wasn't important to remember every details of the proof but if I remembered for example that this theorem follows from Zorn's lemma or by transfinite induction, I was often able to fill the details in the exams. –  Jaakko Seppälä Aug 5 '13 at 16:55
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I don't think I've ever "learned a proof" in the sense of being able to reproduce it word for word. To me, "learning a proof" means understanding the ideas well enough so that (1) I'm sure the result is correct and (2) I could give a proof (presumably related to but not identical with what I learned) if asked. –  Andreas Blass Aug 5 '13 at 17:40
    
Before reading the proof, try to prove it by your self to see what is your obstacles with it . Then figure out what is the main idea behind the proof. –  Ahmed Aug 6 '13 at 0:53
    
@OwenSizemore My exam schedule for this term is very unfortunate. –  user44322 Aug 6 '13 at 6:59

2 Answers 2

up vote 2 down vote accepted

Some tips to learn why theorems are true:

  1. Draw pictures
  2. Come up with examples of the theorem or how to apply it
  3. Come up with counterexamples to illustrate why the theorem fails or its limitations
  4. Try to "summarize" the proof in a few lines.
  5. Figure out the most difficult part of the theorem, and how that difficulty is dealt with.
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This is pretty much what I have been doing. –  user44322 Aug 6 '13 at 7:00

I would typically just advise you to learn the theorems, that is, understand why they are true; this way they are much easier to remember. However, using this method for $23$ theorems that you need to learn in a day is going to be too time consuming...

Your best bet is to understand why the theorems are true, so you can at least make a decent attempt at reproducing the theorem based on definitions while writing your examination.

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