TITLE: Are there proofs that you feel you did not "understand" for a long time?
QUESTION [75 upvotes]: 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.

REPLY [4 votes]: The recent answer by coudy reminds me that for a very long time I didn't really grok the proof of Minkowski's inequality from Hölder's inequality, even though it's fairly simple. It just looks like a clever trick, and not really the first thing one would think of. Put differently, one would like a more conceptual story lurking behind that proof. 
As I understand it now, Minkowski's inequality (or the triangle inequality for the $p$-norm) is really all about convexity of the unit ball. You can prove $L^p$ is locally convex with a little calculus, without Hölder's inequality. But more recently I'd been telling myself the following conceptual story: if the meaning of Hölder's inequality is that there is an isometric isomorphism $L^p \cong (L^q)^\ast$, induced by the standard pairing $\langle -, - \rangle: L^p \times L^q \to \mathbb{C}$, then this implies that the unit ball in $L^p$ is the intersection of sets 
$$H_g = \{f \in L^p: |\langle f, g\rangle| \leq 1\}$$ 
where $g$ ranges over the unit ball in $L^q$. But each $H_g$ is clearly convex, and therefore so is the intersection. The standard proof of Minkowski via Hölder can be seen as just a very tidied up rendition of this more conceptual explanation.