Recently I finished my 4-year undergraduate studies in mathematics. During the four years, I've met all kinds of proofs. Some of them are friendly: they either show you a basic skill in one field or give you a better understanding of concepts and theorems.
However, there are many proofs which seem not so friendly: the only feeling after I read them is "how can one come up with that", "how can such a long proof be constructed" or "why it looks so confusing". What's worse, most of those hard proofs are of those important or famous theorems. All what I can do with these hard proofs is working hard on reciting them, forgetting them after exams and learning nothing from them. That makes me very frustrated.
After failing to find the methodology behind those proofs, I thought, "OK, I may still apply the same skill to other problems." But again, I failed. Those skills look so complicated and sometimes they look problem-specific. And there are so many of them. I just don't know when to apply which one. Also, I simply can't remember all of them.
So my question are: How to learn from those hard proofs? What can we learn from them? What if the skill is problem specific? (How to find the methodology behind them?)
I need your advises. Thank you!
P.S. Threre are a lot of examples. I list only four below.
Proof of Sylow Theorem in Algebra
Proof of Theroem 3.4 in Stein's Real Analysis.
Theorem 3.4 If $F$ is of bounded variation on $[a,b]$, then $F$ is differentiable almost everywhere.
Proof of Schauder fixed point theorem in functional analysis.
Proof of open mapping theorem in functional analysis.