Infinite Product of Analytic Functions

Theorem
Let $D\subset\C$ be an open connected set.

Let $(f_n)$ be a sequence of analytic functions $f_n:D\to\C$ that are not identically zero.

Let $\displaystyle\sum_{n\mathop=1}^\infty(f_n-1)$ converge locally uniformly absolutely on $D$.

Then:
 * $f=\displaystyle\prod_{n\mathop=1}^\infty f_n$ converges locally uniformly absolutely on $D$
 * $f$ is analytic
 * For each $z\in D$, $f_n(z)=0$ for finitely many $n\in\N$
 * For each $z\in D$, $\operatorname{mult}_z(f)=\displaystyle\sum_{n\mathop=1}^\infty\operatorname{mult}_z(f_n)$

where $\operatorname{mult}$ denotes multiplicity.