Let $G$ be a discrete subgroup of $\operatorname{PSL}_{2}(\mathbb{C})$. Show that $G$ is countable. Let $G=\left\{g_{1}, g_{2}, \ldots\right\}$ be some enumeration of the elements of $G$. Show that for any point $p$ in hyperbolic 3-space $\mathbb{H}^{3}$, the distance $d_{h y p}\left(p, g_{n}(p)\right)$ tends to infinity. Deduce that a subgroup $G$ of $\mathrm{PSL}_{2}(\mathbb{C})$ is discrete if and only if it acts properly discontinuously on $\mathbb{H}^{3}$.