Theorems with one-line proofs

Question

Inspired by this very concise answer, which proves that $$\sin^2(\theta)+\cos^2(\theta) \equiv 1 $$ as follows:

$f(\theta)=\cos^2\theta+\sin^2\theta \quad;$

then it's simple to see that $$f'(\theta)=0,$$ then $$f(\theta)=f(0)=1,$$

I'm looking for theorems (and corollaries and lemmata) with one-line proofs.

Rules:

1) The proof of the theorem has to actually deduce something; it can't be, for example, a definition.

2) No vacuous truths; e.g. "If the sky is green, then the Riemann Hypothesis is true" is not allowed.

3) The words "trivially true" are banned!

4) The proof (when rendered in $\LaTeX$), must be under 100 characters, and must be no longer than one sentence.

If you could state the theorem along with its short proof, that would be great!

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Obviously, there are lots of theorems with proofs of varying lengths-- I just care about the short ones!

Answer 1

A famous one : Irrational number to an irrational power can be rational.

Proof : If $\sqrt 2^{\sqrt 2}$ is rational, we are happy. If $\sqrt 2^\sqrt 2$ is irrational, then $(\sqrt 2^{\sqrt 2})^\sqrt 2=2$ is rational.

P.S. $\sqrt 2^\sqrt 2$ is actually irrational because it is transcendental by Gelfond–Schneider theorem, but we don't need to know this theorem to prove the above statement.

Answer 2

Hairy Ball theorem (for $n=2$): There are no non-vanishing continuous tangent vector field for $S^2$

Proof:: If such a vector field did exist, let $v_x$ be the vector at $x$. The function $H:S^2 \times [0,1] \to S^2$ mapping $(x,t)$ to the point $t\pi$ radians away from $x$ along the great circle defined by $v_x$ is a homotopy between the identity and the antipodal map on $S^2$, which is impossible.

Well I wrote two sentences, but essentially it is one.