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 mathematicans should know, and why.
Here is my favourite "wow" proof .  

Theorem
There exist two  positive irrational numbers $s,t$  such that $s^t$ is rational.
Proof
If $\sqrt2^\sqrt 2$ is rational, we may take $s=t=\sqrt 2$ .
 If $\sqrt 2^\sqrt 2$ is irrational , we may take $s=\sqrt 2^\sqrt 2$ and  $t=\sqrt 2$ since  $(\sqrt 2^\sqrt 2)^\sqrt 2=(\sqrt 2)^ 2=2$.
It is propably not something which everybody should know, nevertheless it is simply beautiful!

The stable Hurewicz theorem using that the sphere spectrum is a compact generator of the stable homotopy category.
In particular Serre's theorem that the rational stable homotpy groups of spheres are trivial for degrees bigger than 1.