# Proofs that every mathematician should know. [closed]

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.

There exist two positive irrational numbers $s,t$ such that $s^t$ is rational.
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 $(\sqrt 2^\sqrt 2)^\sqrt 2=(\sqrt 2)^ 2=2$.