I am looking for examples of theorems that may have originally had a clunky, or rather technical, or in some way non-illuminating proof, but that eventually came to have a proof that people consider to be particularly nice. In other words, I'm looking for examples of theorems for which have some early proof for which you'd say "ok that works but I'm sure this could be improved", and then some later proof for which you'd say "YES! That is exactly how you should do it!"
Thanks in advance.
A sister question: Examples of major theorems with very hard proofs that have NOT dramatically improved over time
Best Answer
[Edit: This answer seems to fit the title of the question, though not the actual question in the body.]
Resolution of singularities in algebraic geometry seems like a good example. Hironaka's original proof was over 200 pages and hard to understand:
That quote is from Dan Abramovich's Math Review of the book Lectures on resolution of singularities by Kollár; the review goes on to say
I know almost nothing about this topic, but some names I know associated to the various approaches to simplification of Hironaka's proof are Bierstone, Milman, Encinas, Villamayor, Hauser, Cutkosky, Włodarczyk, Kollár, Cossart, Piltant... Please tell me any I missed!