Bound on the difference of two convergent infinite products

Question

Let $(\alpha_n)$ and $(\beta_n)$ be two sequences of non-zero complex numbers such that the products $\prod_n \alpha_n$ and $\prod_n \beta_n$ are convergent. How to prove the following inequality? $$ \left|\prod_{n\geq 1} \alpha_n - \prod_{n\geq 1}\beta_n\right|\leq\left(\prod_{n\geq 1}\max(|\alpha_n|,|\beta_n|)\right)\sum_{n\geq 1}\frac{|\alpha_n-\beta_n|}{\max(|\alpha_n|,|\beta_n|)}. $$