How to learn from proofs?

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.


One important thing about proofs is that you will never be able to appreciate them, and therefore to learn from them, if you are not capable of reading the statement to be proved with a sceptical attitude, and to try to imagine it is untrue:

  • What’s this nonsense they are claiming, it cannot be true!

  • Certainly it must be possible to satisfy the hypotheses without being obliged to accept the conclusion!

Once you have some mental idea of what a counterexample to the statement would look like, you can interpret the proof as an argument that systematically talks this idea out of your head, convincing you that it really is not possible to ever come up with such a counterexample.

Then you will have acquired a feeling of what the proof is really about, and you will be far more likely to retain it, and to come up with similar arguments when you need to prove something yourself.

But if you take a docile attitude and accept the statement to be proved from the onset, you will never be able to understand what all this reasoning was needed for in the first place.

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