How to get better at Proofs

Question

I'm still quite early in my Journey of Mathematics and as I have progressed I have found a constant wall; proofs. Every time I try and learn to do simple proofs, I struggle to find an actual strategy, and locating what I am actually trying to prove. I feel as if that when I construct a proof, I try and reuse exactly what the textbook has told me, but in the end I still get the proof wrong, because of not noticing a small detail or not managing to construct my OWN proof. I am wondering if there is a general strategy, or way to learn to just construct your own proofs without relying on outward sources.

Answer 1

First of all, to learn how to produce proofs one must learn how to read proofs. Say you're studying real analysis and you really struggle with $\epsilon-\delta$ proofs. This is a common occurrence. What makes you better at it? First, try to see a lot of examples. Each mathematical discipline has a certain flavor and so do the proofs of the results in that subject. So read a lot of proofs and go through many examples in the theory. That is the first step. Next, before you actually read a proof, try to think of what the person is trying to prove. Is it reasonable? Do you have an idea that might help you prove it? Read the proof. Is it what you expected? Did they use all the hypotheses? Where were they used? Keep asking yourself these questions and things will get easier. Lastly, try to revisit proofs yourself and try to prove more basic results on your own. Do a lot of exercises. This is the hardest part of studying mathematics in my opinion: gaining mathematical maturity.

Answer 2

Different take: Dont focus of proofs. Focus on solving problems. Then try to explain your solutions as clearly, concisely and rigorously as possible.