Let $(f_n{)}_{n\in \mathbb{N}}:G\to \mathbb{C}-(-\mathrm{\infty },0]$ be a sequence of holomorphic functions, with $G$ a domain. Then $\prod _{n=1}^{\mathrm{\infty }}{f}_n(z)$ converges locally uniformly iff 1;;$\sum _{n=1}^{\mathrm{\infty }}\ln (f_n(z))$ converges locally uniformly, of course with the arguments of the logarithms chosen carefully

Proof:

1. We essentially use 5. Proposition 2.12.1 to talk about pointwise convergence of the product, and extend it to local uniform convergence.