The following theorem appears in "Princeton Lectures in Analysis, Complex Analysis," by Stein & Shakarchi:
Suppose $\{F_n\}$ is a sequence of holomorphic functions on the open set $\Omega$.
If there exists constants $c_n \gt 0$ s.t. $\sum c_n < \infty$ and $|F_n(z) - 1| \le c_n$, for all $z \in \Omega$, then
The product $\prod_{n=1}^\infty F_n(z)$ converges uniformly in $\Omega$ to a holomorphic function $F(z)$
The idea is that we can write $\prod_{n=1}^N F_n(z)$ as $\prod_{n=1}^Ne^{\text{log}F_n(z)} = e^{\sum_{n=1}^N\text{log}F_n(z)}$
Using the fact that $|\text{log} f| \le 2|f-1|$ for $f$ close enough to $1$, together with $|F_n(z) - 1| \le c_n$ and $\sum c_n < \infty$, it follows that $e^{\sum_{n=1}^N\text{log}F_n(z)}$ converges to some function $F(z)$,
but how do we know that it converges uniformly?
Hint: bound $\left|\sum_{n=M}^N \log F_n(z)\right|$