Prove that this complex series converges uniformly

Question

Let $\{a_n \}_{n=1}^\infty$ be a sequence of complex numbers, so that $0<|a_n|<1$, and it verifies $\sum_{n=1}^\infty(1-|a_n|)<\infty$. Prove that the series $$\sum_{n=1}^\infty\left( 1-\frac{|a_n|}{a_n}\frac{z-a_n}{\bar{a}_nz-1} \right) $$ converges uniformly in the compact sets of the unit disc (just need to prove that it converges uniformly when $z\in B(0,r), 0<r<1$). I have found a closely related question: Prove that this infinite product converges uniformly,; although what the author wants to prove in the linked question needs what I'm asking for here. I've tried to manipulate the series term using the bound $|z|<r$, but I've got nothing. Any hints would be highly appreciated.

Answer 1

$|1-\frac{|a_n|}{a_n}\frac{z-a_n}{\bar{a}_nz-1}|=|\frac{(a_n+|a_n|z)(1-|a_n|)}{a_n(1-\bar a_n z)}|$

Since $|\frac{a_n+|a_n|z}{a_n(1-\bar a_n z)}| \le \frac{2}{1-r}, |z| \le r$ the conclusion follows from the convergence of $\sum_{n=1}^\infty(1-|a_n|)$