Perhaps the "proofs" of ABC conjecture or newly released weak version of twin prime conjecture or alike readily come to your mind. These are not the proofs I am looking for. Indeed my question was inspired by some other posts seeking for a hint to understand a certain more or less well-establised proof, or some answers to those posts. I am interested in proofs at undergraduate levels. Since the question as asked in the title would be too personal, I suggest a longer and hopefully more positive version:
Is there any proof that you feel you didn't understand fully until years later?
The reason that I am interested in this question is that we are currently working on an assessment framework for assessing students' understanding of proof. Reading some previous posts on MO, It occurred to me that perhaps we are too naive in our approach just seeking for understanding logical structure, the key point and so on. It would be very informative if you kindly include in your answer the follow-up of the proof you mention.
Best Answer
As an undergraduate, I learned the Sylow theorems in my algebra classes but could never retain either the statement or proof of these theorems in memory except for short periods of time (and in particular, for the duration of an algebra exam). I think the problem was that I was exposed to these theorems long before I had internalised the concept of a group action. But once one has the mindset to approach a mathematical object $X$ through the various natural group actions on that object, and then look at the various dynamical features of that action (orbits, stabilisers, quotients, etc.) then all the Sylow theorems (and Cauchy's theorem, Lagrange's theorem, etc.) all boil down to observing some natural action on some natural space (e.g. the conjugacy action on the group, or on tuples of elements on that group) and counting orbits and stabilisers (p-adically, in the case of the Sylow theorems). (Isaacs book on finite group theory emphasises this perspective very nicely, by the way.)