I'm currently taking a Transition to Advanced Mathematics course, which is entirely proof-based, so it's pretty new territory. Up until now, all the classes I've taken were fairly computational, so studying was just doing practice problems. I've already made flash cards of all the definitions (which includes stuff from elementary set theory like Cartesian products, cardinality, countable sets, etc.), but it just doesn't feel sufficient to know the definitions. Should I try to prove everything I can think of/find in the book? For those who have done it before, what did you do to study for a proof-based final and was it successful?
Yes, generally everyone's first encounter with writing proofs is a very stressful time because really there's no way to memorize a few steps to get you through problems. My advice is just practice as much as you can, do all of the practice problems in your book, ask your professor for problems relevant to your course, etc. Another thing to make sure is that your proofs are clear and logically correct. By this I me an assume the correct things, state your definitions precisely and don't be afraid to leave out details. I remember writing book reports in grade school and the teacher would always say "write as if I've never read the book before", same kind of principle, write your proofs as if the person reading knows nothing about math (for this intro proofs class anyways). If you have time, get your prof to go over some proofs that you've written and ask for constructive criticism, or even post them here and we would be happy to help. Good luck!