Hello econ PhD folks! I'd like to introduce myself and possibly obtain some advice from such wizened mathematicians as yourself. I'm an MSc student studying economics at a top terminal MSc program in the US. During my next semester, I'll be taking my university's graduate micro course and its math accompaniment (which covers some topics in analysis, topology, and nonlinear optimization), as well as a proper introduction to topology course. In preparation, I'm trying to get through chapters 1-4 of the Principles to Mathematical Analysis book by Walter Rudin to make life easier in the future.
However, (surprise, surprise) I am finding this brutally difficult. I've been working through the first chapter's exercises, and I've had to spend hours upon hours to make any headway. For a few of the recent problems (6a-d particularly, if anyone is familiar with the book), I've not been able to really figure out an approach, and so I'll just gradually expose myself to a solution and see line-by-line if something sparks an understanding.
Does anyone with a bit more experience have possible suggestions for how to think about this stuff and what to do if you're trying to teach yourself stuff like this? Also, I considered posting this in r/learnmath, but that seems to be mostly focused on calc and earlier classes.