entire function with bounded multiplicity is a polynomial

Question

Let $f:\mathbb{C}\to\mathbb{C}$ be an entire function.

Let $n\in\mathbb{N}$ and suppose that

$$\forall w\in\mathbb{C}:\#\{z\in\mathbb{C}:f(z)=w\}\leq n$$

In words, every complex value is attained by $f$ in at most $n$ different places.

Prove that $f$ is a polynomial of degree at most $n$.

Answer 1

This follows from Picard's great theorem.