I looked at rudin and some others and I feel like Rudin's real analysis tests you on your ability to pull magic tricks out of a black hat rather than on your ability to know key definitions,corollaries. What's a book that focuses on your ability to know definitions, corollaries well, that isn't Euclid's elements that introduces you to doing proof that will actually help you do upper level math? I looked at proofwiki.com and I'm wondering whether knowing their correct definitions/proofs on there actually helps one do real analysis or upper level math.
Can't promise this is a particularly good idea (you only get to introduce yourself to proofs once) - but I did the above with Lang's "Linear Algebra". Before that, I took "introduction to Mathematical thinking" on Coursera. Not a textbook, but the course was very informative and helped me get a grasp of proof writing, as well as some traps to avoid.
Proofwiki seems like an inefficient way to go about this. If you knew and understood everything there, good. You'll be ready for many many things. But I'm not sure it'd be presented in a particularly efficient way, nor is it the best place to actually practice these things for yourself.