TITLE: Should one learn the proofs of theorems which have highly complicated proofs?
QUESTION [8 upvotes]: Lately, I have been reading a really dense real analysis textbook and I came across different theorems which have exceedingly long proofs (for instance, the dominated/bounded convergence theorem or many other theorems that involve interchanging the order of integration/differentiation).
I wonder if it is worth learning such proofs, because, for instance, I have known the DCT for a while and I have used it extensively to compute different limits, but I can't see why it would be useful to know how to prove it. I would like to add that I am mostly self-learnt at this level, because I have just finished high school. As a result, I don't know if, for instance, in a college level real analysis course the lecturer would prove such a result.
I chose the DCT as an example because I have found it quite useful in different problems, but there are many other theorems which fit into this category.

REPLY [2 votes]: "Analysis" literally means "breaking something complex into simple pieces". Attacking anything too complicated directly will usually just give you a bruise from running into a wall. That applies at scales from big open problems, to reading a paper, to textbook exercises. The solution is almost always to just steadily pick away at the edges. There's a famous Grothendieck quote about this (McLarty's translation):

I can illustrate the ... approach with the ... image of a nut to be opened. The first analogy that came to my mind is of immersing the nut in some softening liquid, and why not simply water? From time to time you rub so the liquid penetrates better, and otherwise you let time pass. The shell becomes more flexible through weeks and months — when the time is ripe, hand pressure is enough, the shell opens like a perfectly ripened avocado!

If you don't think they're getting much from reading "highly complicated proofs", it probably means you haven't broken things into enough simple pieces. Trying to pick out the intuition behind the proof is one approach. Moving on to exercises and coming back to the proof later when you're stuck is another.