Proof:

Suppose an entire function $f$ misses two points $p$ and $q$. The map $f:\mathbb{C} \longrightarrow \mathbb{C} - \{p, q\}$ lifts to a map ${\tilde f}:\mathbb{C} \longrightarrow \{z\in \mathbb{C}/ \vert z\vert < 1\}$. As before the lift is holomorphic and hence is an entire function taking its values in the unit disc. By Liouville's theorem, ${\tilde f}$ is constant and so must $f$.