I am studying Real Analysis -- using Baby Rudin -- on my own in a community that does not have a university near (if the "why" is important, see below). I recently worked through standard texts for Calculus, (basic) Differential Equations, Linear Algebra, Vector Calculus and [Eccles] "Introduction to Mathematical Reasoning" in preparation. Most of those were fairly easy to do in a self-study environment, because it was reasonably easy to tell, when doing the problem-sets, whether I'd gotten it right or not. (Student & Instructor answer books and Chegg helped a good bit when I got stuck; which was not often, but more often than I'd like .)
However: while I have the answer-book for Rudin, I'm finding it much harder to tell -- where most problems are of the form "Prove X" -- whether my proofs are valid unless they match precisely those in the answers. Sometimes they do. Usually there are subtle (or greater) differences. And I cannot reliably tell whether those 'subtle differences' are inconsequential or horrifyingly stupid (plus WRONG). Anyone with thoughts or experience on how this subject can best be learned in a self-study environment? Or is it likely I'm going to have to develop a resource -- one way or the other -- to 'grade' my efforts and help keep me on track? Thanks for considering.
[Possibly unnecessary framing information: Mostly I just want to spiff up my mathematical understanding. But there exists a research project that I'd like to join, in my field (Medicine), which really needs at least some ability to think topologically about the problems, and which often involve (somewhat) the math of Chaos. Thus I think I need to bring myself up to at least that level to participate.]