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It’s often said that, when you see a new theorem in a textbook, you should first try to prove it. And Bellman said that, if you can solve a problem, it’s an exercise; if you can’t, it’s a research problem. My question is: If you can’t prove a textbook theorem for a long time, what should you do? Should you give up and move on, or should you keep working on it?

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closed as primarily opinion-based by Cameron Williams, Lee David Chung Lin, cmk, metamorphy, TheSimpliFire Jun 23 at 19:45

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ You should ask for a hint on SE. $\endgroup$ – Ruben Jun 23 at 11:28
  • $\begingroup$ I think this is a difficult calibration question. I don't know how to tune that parameter optimally, and I wonder if some people got lucky in that their natural inclinations happen to be for a highly efficient tradeoff between trying to work through details on one's own and quickly getting a big picture overview of a large amount of material. Maybe the best approach is to be motivated by beauty, so that reading a math book is like reading a poem or a comic book -- you just read in whatever greedy/irreverent way feels most fun, and revisit the material later if you need to understand it better. $\endgroup$ – littleO Jun 23 at 11:35
  • $\begingroup$ I think it greatly depends on the Theorem. Some results, like the Prime Number Theorem or the Poincare Conjecture, are so difficult that it really wouldn't be sensible to put everything on hold while you attempted to solve them on your own. Doesn't mean it's not useful to try for a little while though...get a feeling for what the Theorem says and, at least, convince yourself that some approaches won't yield easy proofs. $\endgroup$ – lulu Jun 23 at 11:37
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    $\begingroup$ Study should be a parallel process, not a sequential one. If one cannot do an exercise/prove a theorem/understand a concept. Take a note at it and move on. Revisit it again later. If you still don't know how to deal with that. Make another note, move on and revisit it again later. If you can't deal with it the third time. Make a note and forget about it.... $\endgroup$ – achille hui Jun 23 at 11:52
  • $\begingroup$ Depends on how important the problem is to you. (There are many, many reasons why we attempt proofs. Sometimes, it's because of a penalty if we don't---which might be the most likely reason for "textbook'' problems.) $\endgroup$ – mlchristians Jun 23 at 17:49
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That depends on the theorem. There are many kinds of theorems in textbooks:

  1. Theorems introduced with "it is now easy to prove ..." but no actual proof.

    Here it is expected that you will be be able to find the proof without too much struggle. If you can't, it's generally a sign that you haven't internalized the preceding material well enough to proceed yet, or that you're too inexperienced with proofs in general to follow the pace of the book, or that it's a typo, or that the book is just bad.

    Continuing to bang your head against the wall with it indefinitely might not be the best cause of action. Other possible plans include asking here, or looking for a different book that covers the subject.

  2. Theorems that the book states and then proves.

    It is often a good idea to attempt to prove such a theorem from just the statement of it. Even if unsuccessful this ought to give you some understanding of what it is the theorem claims and what the obstacles in proving it are, and you will then be in a better position to appreciate the central idea in the proof that follows.

    Since the the book deigns to spend ink on the proof, it is not expected that you will necessarily succeed in finding a proof. After you have spent some time investigating attacks if you still don't find any, go on and read the book's proof instead.

  3. Theorems that the book mentions in passing, but without proof. ("Sutch and Such (2003) have shown that the converse implication holds in general iff a weakly hyper-ambivalent cardinal exists").

    Readers are generally not expected to be able to reconstruct those proofs.

  4. Things you're asked to prove in exercises generally lie somewhere between class 1 and 2. Some books will try to mark a distinction about exercises that are expected to be easy to prove, and "bonus exercises" that could take a good student days to crack. Not all do.

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  • $\begingroup$ Thanks for your answer! One problem holding me stuck is Bernstein’s theorem, stating roughly that if A maps to a subset of B and B maps to a subset of A then A maps to the entirety of B. It’s not required and proved in small font. But I was afraid that if I gave up on that too easily, I would lose out when doing research later on. A genius should be able to prove everything, right? I’m not of course. $\endgroup$ – Zirui Wang Jun 24 at 7:26

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