Deepest theorems with simplest proofs

Question

Which are the deepest theorems with the most elementary proofs?
I give two examples:
i) Proof_of_the_Euler_product_formula_for_the_Riemann_zeta_function
ii) Proof that the halting problem is undecidable using diagonalization

Answer 1

These perhaps aren't particularly deep, but they are the first that come to mind.

1. Irrationality of $\sqrt{2}$ by contradiction.
2. Uncountability of the reals by diagonalization.
3. Existence of graphs with arbitrarily high girth and chromatic number by the probabilistic method.

Answer 2

I think one should not confuse "important with "deep". The facts that $\sqrt{2}$ is irrational, that there is no surjective map $X\to2^X$, or that there are an infinity of primes, are certainly important or even "fundamental", but their proofs are so simple that one cannot call them "deep". A theorem is "deep" when its proof is really hard and, above all, requires a theory that transcends the realm the problem is formulated in. Consider, e.g., Gauss' theorem about which regular $n$-gons can be constructed with ruler and compass.