Convergence of the infinite product $\prod_{n = 1}^{\infty} \frac{z - \alpha_n}{z - \beta_n}$

Hint:

Suppose that $z\in\mathbb{C}\setminus\overline{\{b_n\}}$, then there is an $\epsilon>0$ so that $|z-b_n|\ge\epsilon$ for all $n$. Then, $$ \sum_n\left|\frac{b_n-a_n}{z-b_n}\right|\le\frac{1}{\epsilon}\sum_n|b_n-a_n|<\infty\tag{1} $$ Inequality $(1)$ implies that $$ \prod_n\frac{z-a_n}{z-b_n}=\prod_n\left(1+\frac{b_n-a_n}{z-b_n}\right)\tag{2} $$ converges.