"Simple" beautiful math proof Is there a "simple" mathematical proof that is fully understandable by a 1st year university student that impressed you because it is beautiful?
Here's a cute and lovely theorem.


  There exist two irrational numbers $x,y$ such that $x^y$ is rational.


Proof. If $x=y=\sqrt2$ is an example, then we are done; otherwise $\sqrt2^{\sqrt2}$ is irrational, in which case taking $x=\sqrt2^{\sqrt2}$ and $y=\sqrt2$ gives us: $$\left(\sqrt2^{\sqrt2}\right)^{\sqrt2}=\sqrt2^{\sqrt2\sqrt2}=\sqrt2^2=2.\qquad\square$$

(Nowadays, using the Gelfond–Schneider theorem we know that $\sqrt2^{\sqrt2}$ is irrational, and in fact transcendental. But the above proof, of course, doesn't care for that.)
I like the proof that not every real number can be written in the form $a e + b \pi$ for some integers $a$ and $b$.  I know it's almost trivial in one way; but in another way it is kind of deep.