Liouville's theorem

Statement

Let $f : \mathbb{C} \to \mathbb{C}$ be an entire function. Assume that $f$ is bounded: there is a real number $M$ with $|f(z)| \le M$ for each $z \in \mathbb{C}$. Then $f$ is constant.

Remarks

Proof: let $c \in \mathbb{C}$. By Cauchy's estimates, $|f'(c)| \le M / r$ for each $r > 0$. So $f'(c) = 0$ for each $c$, and $f$ is constant. For example, $\sin$ is entire and not constant, so it is not bounded on $\mathbb{C}$. On $\mathbb{R}$ it is bounded, which shows that the theorem has no real analogue. The theorem gives a short proof of the fundamental theorem of algebra.

Explanations

L1 human

A holomorphic function on the whole plane cannot stay inside a bounded region unless it does not move at all. Each nonconstant entire function, for example $z$, $e^z$ or $\cos z$, takes values that are as large as you like.