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.