A question that comes up a lot in class is, “how do you get good at proofs?” (It’s usually framed as “I don’t like proofs”, but we’re not having any of that negativity here, thank you very much.)
I don’t have a silver bullet for that. I do have some advice.
In your special proofs notebook, keep track of all the tricks that seem obvious in retrospect - or that make you say “I’d never have thought of that!” When you’re stuck, flick back through it and see if anything jumps out as a possible way forward.
I can’t explain the process of walking - when I was a kid, I just sort of kept falling over until I got passably good at it. Similar thing with proofs: try. Fall over. Try again.
The more you do, the easier it gets.
Try to get a feel for the kinds of problems that tend to use one method or another. After a while, you develop a sense of smell for it.
I find that many proofs have a “nugget” - a key flash of insight in the middle of routine work. Focus on those nuggets. Respond to them with an ‘oo!’.
If a question gives you details - for example, that the function isn’t defined when $x=a$, or that your integer is odd, or similar, ask questions like:
Proofs are full of all sorts of rules about keeping things equivalent and what you can or can’t do. But those are for presenting proofs. When you’re trying to work out the way to a proof, it’s absolutely fine to disregard those rules and focus on convincing yourself.
You can tighten up the presentation later - but playing around is usually how you get to the nugget.
Do you have any top tips for generating proofs? I’d love to hear them in the comments.