Let $\Omega \subset \mathbb{C}$ and take $\{ f_n \}$ to be a sequence of conformal mappings on $\Omega$. Suppose that there is some $z_0 \in \Omega$ such that $f_n(z_0) \to \omega \in \partial \Omega$. I want to prove that there is a subsequence of $\{ f_n \}$ that converges to $\omega$ in $\mathscr{H}(\Omega)$.
I know that if the $f_n$ are conformal then the limit in $\mathscr{H}(\Omega)$ is conformal also, but this problem is really hard.
We can perhaps use Arzela-Ascoli, since the holomorphic functions are on a bounded set, therefore bounded? Maybe Montel's theorem?
I think it is done with Hurwitz's theorem applied to the holomorphic function $f(z) - \omega$.