Proofs that every mathematician should know. 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 mathematicians should know, and why.

I really like the simple and nice proof of the 5-color theorem (i.e. that for every planar graph there exists a vertex coloring with not more than 5 colors) and how surprisingly difficult it is to proof the sharper 4-color theorem.
