I think the following statement is true but I'm not completely sure.
Let $\Omega$ be an open set of the complex plane and let $(f_n)_{n=1}^\infty$ be a sequence of holomorphic functions defined on $\Omega.$ If the series $$\sum_{n=1}^\infty |1-f_n|$$ converges uniformly on compact subsets of $\Omega,$ then the infinite product $$\prod_{n=1}^\infty f_n$$ converges absolutely on $\Omega,$ uniformly on compact subsets, and the function $f$ it defines on $\Omega$ is holomorphic. In particular $f(z) = 0$ if and only if $f_n(z) = 0$ for some $n \in \mathbb{N}.$