I need to show that $\Pi \frac{|a_n|}{a_n} \frac{a_n - z}{1-\overline{a_n}z}$ with $\{a_n\} \subset D(0,1)$ and $\Sigma$ $(1 -|a_n|) < \infty$ converges to a holomorphic function in $D(0,1)$
It's clear that each $\frac{a_n - z}{1-\overline{a_n}z}=\varphi_{a_{n}}$ with $\varphi_{a_{n}}$ an automorphism of the unit disk.
Each term of the infinite product is bounded and $|a_n|$ converges to 1 because $\Sigma$ $(1 -|a_n|) < \infty$
I'm trying to use this:
Let $\{f_n \}$ be a sequence of holomorphic functions in an open set of the complex plane ($\Omega$) if
$\Sigma$ $|f_n - 1 |$ converges uniformly on the compact sets of $\Omega$ then $\Pi \{f_n \}$ also converges.
The problem is that i'm not finding anything useful with the manipulations of $|f_n -1|$. Any advice?