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The proofs presented in lectures, textbooks ect. are usually cleaned up versions that show just the necessary steps for logically proving the theorem, not the thought process that went into the proof.

To give a concrete example, I'm working through an (overall quite good) MIT Open Course Ware class on real analysis. I just paused a lecture where the professor said:

Now, when you write a proof, as you'll see, it's going to be magic that somehow this h does something magical. That's not exactly how you come up with proofs. How it comes up is you take an inequality that you want to mess with, you fiddle around with it, and you see that if h is given by something, then it breaks the inequality or it satisfies the inequality, which whichever one you're trying to do.

And proceeded to just write the finalized proof. Because that's not the part I care about. The main thing I want to learn is the part were you do the fiddling around to come up with the proof to begin with. Verifying proofs other people made is (relatively) straightforward, and the property being proven (Q doesn't have the least upper bound property) is important but I'd be willing to take it as an assertion if I was just trying to learn about Q rather than how to do analytical proofs.

I would really like to see examples of someone who is good at proofs showing their work in creating a new one including the dead ends, fiddling around ect.

I have tried to teach myself this step by just going out and proving things, and have made a little bit of progress but thing I would greatly benefit from more examples of deriving proofs.

Hunter
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  • i had this exact question once, and you will learn this many ways. One, literally fiddle around with whatever you’ve got. Even if you end up with nonsense in the end, it gets your brain juices flowing. Then, you work out someone else’s proof. Then, and here comes the most important part, try to figure out if there’s some way you can ‘recreate’ their proof (it doesn’t have to be an identical match, and often in analysis it won’t, but see if you can try to get close). – peek-a-boo Jul 30 '23 at 07:44
  • note the reason people don’t show you the gory details of the thought process is that often it is not important (sometimes it can be instructive, but other times, it’s not) and often it can be way too long, and sometimes inefficient. This can be disheartening because you’re left wondering how the heck anyone came up with such an idea, but that’s just because (aside from the possibility of being a genius) they just have more experience, so they ‘know’/have a feel for what to try and what not to try. – peek-a-boo Jul 30 '23 at 07:46
  • By very long-standing tradition, proofs in papers and textbooks are cleaned up. This is partly for reasons of space, partly from the feeling expressed by Gauss’s motto, “Few, but ripe.” That said, I know of two papers that broke with this tradition: “How not to prove Fermat’s Last Theorem”, and “How Not To Prove The Poincaré Conjecture”. – Michael Weiss Jul 30 '23 at 17:22
  • You may also find something useful in this answer to the question about “prove or find counterexample” problems: https://math.stackexchange.com/questions/4657136/improve-at-proof-or-find-counterexample-exercises/4657500#4657500 – Michael Weiss Jul 30 '23 at 17:28
  • Thanks for the responses. – Hunter Aug 03 '23 at 10:25

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