TITLE: How to learn from proofs?
QUESTION [102 upvotes]: Recently I finished my 4-year undergraduate studies in mathematics. During the four years, I 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 I have after reading them is "how can one come up with that", "how can such a long proof be constructed" or "why does it look so confusing". What's worse, most of those hard proofs are of those important or famous theorems. All that I can do with these hard proofs is work hard on reciting them, leading me to forget them after exams and learn nothing from them. This 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 questions are: How to learn from those hard proofs? What can we learn from them? What if the skill is problem-specific? (How do I find the methodology behind them?)
I need your advice. 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.

REPLY [15 votes]: I find it very helpful to revisit proofs after some time. I find there's two levels of understanding a proof:
--> line-by-line: you find the proof written in a particular textbook. for each sentence in the proof, you understand how it follows from the prior one.
--> conceptually: you can reproduce the proof because at each step it's clear to you what to do. (maybe there's a calculation or two that involves a random trick that you have to look up.)
Obviously the goal is to reach the second arrow.
Reading a proof over and over until you have reached the first arrow is sort of like doing an exercise, and, like all exercises, it helps acclimate your mind to the ideas at hand, and, in this way, it helps you get to the second arrow. But there's also an essential ingredient of time. Come back in three months, see how you feel.
You might think the second arrow entails the first, but actually it can still be hard to read an author's particular way of writing a proof, even when you understand the proof of a theorem. For instance, I think I could write down the proof of the Baire category theorem (don't quiz me), but if I were to go read a proof in a book, I'd probably get confused about the particulars of the notation without spending a little time, or, worse, convince myself that I was following. "You just do the obvious thing right? Yeah, ok, so wait, why is he saying $x_n \in B_n$, wait, what's $B_n$, oh, whatever, I know this."