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?
closed as primarily opinion-based by Cameron Williams, Lee David Chung Lin, cmk, metamorphy, TheSimpliFire Jun 23 at 19:45
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That depends on the theorem. There are many kinds of theorems in textbooks:
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.
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.
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.
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.