Liouville's Theorem (Complex Analysis)

Theorem
Let $f: \C \to \C$ be a bounded entire function.

Then $f$ is constant.

Remark
In fact, the proof shows that, for a nonconstant entire function $f$, the maximum modulus $\ds \map M {r, f} := \max_{\cmod z \mathop = r} \cmod {\map f z}$ grows at least linearly in $r$.