I have been studying (real and complex) analysis lately, both independently and part of my college courses. While I don't have that much of a problem following thee theorems (I can prove say most of them by myself without seeing the book's proof) or with the exercises, I don't possess a big picture of what I'm learning.
When I am reading a theorem with multiple lemmas and the theorem is pretty long and/or tricky, I can locally understand the proof of the lemmas and how one lemma follows from the other but I get so much bogged down in the details that I completely lost track of the big picture/intuition regarding the theorem ! I also lose track of the obstacles of the proof and how they're overcome (rather my attention shifts on understanding the lemmas locally).
Among the theorems which are relatively short and whom I can at least have an intuition just about how the theorem is proved, I see don't really understand how the proof fits into the natural context of the theory or how the proof techniques generalize to other problems (eg when I use this seemingly random trick used in this proof to solve some other classes of problems ?) ?
How can I approach analysis so that I get at least some sense of the big picture and stuff and don't get completely lost in the details?