TITLE: Proofs that every mathematician should know.
QUESTION [443 upvotes]: There are mathematical proofs that have that "wow" factor in being elegant, simplifying one's view of mathematics, lifting one's perception into the light of knowledge, etc.
So I'd like to know what mathematical proofs you've come across that you think other mathematicans should know, and why.

REPLY [13 votes]: My all time favorite proof? 
Furstenberg's proof of the infinity of primes by point-set topology. I know, a lot people think it's not a big deal. I think: 
a) It's an immensely clever way to use point set topology to prove a result in a seemingly unrelated field, namely number theory. 
b) I used it as the beginnings of my first research in additive number theory; looking to generalize this result to create similar proofs of results for sumsets and arithmetic progressions. Sadly, my health failed again,but I hope to return to this research soon.