meromorphic functions

Let $E$ be a compact connected subset of the complex plane that contains more than one point. Show that the family of meromorphic functions on a domain $D$ that omits $E$ (that is, with range in $\mathbb{C}^* \backslash E)$ is a normal family of meromorphic functions.

The hint in the back of the book says to map $\mathbb{C}^* \backslash E$ conformally onto $\mathbb{D}$ , apply thesis version of Montel's theorem. That version states:

Suppose $\mathcal{F}$ is a family of analytic functions on a domain $D$ such that $\mathcal{F}$ is uniformly bounded on each compact subset of $D$ . Then every sequence in $\mathcal{F}$ has a subsequence that converges normally in $D$ , that is, uniformly on each compact subset of $D$ .

I am still confused on how to prove this. I would appreciate some help on this problem. Thanks.