With the current situation involving the novel coronavirus moving my university to remote learning, both of my proof based math classes have been essentially turned into reading courses. This is my first semester taking proof based math courses and I have never had to teach myself this type of material from the book. So I was wondering if anyone has any advice on the most effective ways to read my textbooks. I am studying introductory real analysis from Pugh's Real Mathematical Analysis and Elementary Number Theory from Levesque's Fundamentals of Number Theory. I have seen some people suggest working through each proof on my own before reading the ones in the book, however this seems incredibly inefficient. Would it be just as advantageous to work through the proofs line by line and understand what was done that way? Any advice on this would be very helpful since being able to effectively read and understand mathematics text will be an invaluable skill for me in the future. TIA.

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    $\begingroup$ Pugh’s analysis is the hardest analysis book . Royden’s analysis is the hardest graduate level among all book at same level. $\endgroup$
    – DeepSea
    Mar 13, 2020 at 22:27
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    $\begingroup$ My advice: Whenever you see a theorem statement, stop and think for at least a couple minutes about how a proof would go and see if you can prove it yourself. Ideally you would prove everything yourself, but with limited time that may not be practical. $\endgroup$ Mar 13, 2020 at 22:28
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    $\begingroup$ I also spend some time attempting to prove the theorem myself before I read the proof. When I read a proof, I might only read part of it at first, and then attempt to finish the proof myself once I see the idea. After I've read the proof, I often attempt to reproduce the proof myself from scratch. And I'm always trying to see what is the key idea or technique behind the proof, and is there a nice viewpoint that makes the proof seem easy or obvious in hindsight. Can I imagine a thought process that would lead someone to naturally discover this proof? You have to really grapple with it. $\endgroup$
    – littleO
    Mar 13, 2020 at 22:37

2 Answers 2


Don't read mathematics like a novel. Read it by layers:

  1. first layer : look at theorems, and look backwards at definitions as needed to understand the statement of the theorems. Focus only on statements and ignore the proofs. try to figure out which theorems are most useful, and try to think about which type of problem they could help solve.

  2. second layer : read again more carefully all definitions, propositions and theorems. try to get a feel of the general flow of the chapter.

  3. third layer : for every definition, try to come up with examples and counter examples (for instance, if you are reading about continuous functions, come up with examples of continuous functions, and examples of discontinuous functions). try to match intuition with the formal definition. for every theorem, try to figure out if you find the statement intuitively obvious or, at the opposite, counter-intuitive. try to come up with either a proof or a counter example (you shouldn't succeed). have a look at a few simple proofs, and a quick look at difficult proofs.

  4. lastly, read the remaining proofs in detail.

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    $\begingroup$ I like this outline but would add a different third layer for each theorem: try to understand the simpler corollaries and other applications of a theorem (as they might be presented in the textbook). Seeing how a theorem is applied will enchance your understanding of the theorem. $\endgroup$
    – Lee Mosher
    Mar 13, 2020 at 22:58

One of the best pieces of "how to read a textbook" advice I've ever received is: Read the questions in a chapter FIRST. Then you will know what is important, and how you'll be asked about them later.

As for proof books in particular.... I would recommend you understand the proofs as thoroughly as you can, and then at least try to find alternative proofs, or new counter-examples. When reviewing, go through the theorems and be able to state the proof method for each, even if you don't reproduce the full proofs. This will give you a high-level understanding that will hold you in good stead on exams and years later, when the specifics of proofs elude you (but can be quickly looked up).

In a different mathematical realm... My best preparation for calculus exams was reading through every integral problem in the book and saying: "Oh... because of that ratio, integration by parts should work; because of that difficult square root, a trig substitution should work; because this can be converted into a Gaussian form, the famous conversion from rectilinear-to-radial method should work"... and so on... even if I didn't actually perform those integrals.

Hope this helps!


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