Uniform convergence of $f(z)=\prod_{n=1}^{\infty}\dfrac{\alpha_n-z}{1-\bar{\alpha_n}z}\dfrac{|\alpha_n|}{\alpha_n}$

Question

Let $\{ \alpha_n\}$ be a sequence in the unit disc $\mathbb{D}$ such that $\alpha_n \neq0$ for all $n$ and $\sum_{n=1}^{\infty}(1-|\alpha_n|)$ converges. Show that for each $0<r<1$, the product $$f(z)=\prod_{n=1}^{\infty}\dfrac{\alpha_n-z}{1-\bar{\alpha_n}z}\dfrac{|\alpha_n|}{\alpha_n}$$ converges uniformly for $|z|\le r$.

I can't figure out even if the given product converges pointwise, since the given product may oscillate. Also, how should I prove the 'uniform' convergence for every closed disc?

Answer 1

Hint: $$\frac{\alpha_n-z}{1-\bar{\alpha_n}z}\frac{|\alpha_n|}{\alpha_n}=\frac1{|\alpha_n|}\left(1-\frac{1-|\alpha_n|^2}{1-\bar{\alpha_n}z}\right)$$