Zeroes of Infinite Product of Analytic Functions

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

Let $\left\langle{f_n}\right\rangle$ be a sequence of analytic functions $f_n: D \to \C$.

Let $\displaystyle \prod_{n \mathop = 1}^\infty f_n$ converge locally uniformly to $f$.

Let $z_0\in D$.

Then:
 * $(1): \quad$ $f$ is identically zero some $f_n$ is identically zero
 * $(2): \quad$ $f_n \left({z_0}\right) = 0$ for finitely many $n \in \N$
 * $(3): \quad$ If $f$ is not identically zero, $\operatorname {mult}_{z_0} \left({f}\right) = \displaystyle \sum_{n \mathop = 1}^\infty \operatorname{mult}_{z_0} \left({f_n}\right)$

where $\operatorname{mult}$ denotes multiplicity.

Proof
Note that by Infinite Product of Analytic Functions is Analytic, $f$ is analytic.

Let $n_0 \in \N$ and $U\subset D$ an open neighborhood of $z_0$ such that $\displaystyle \prod_{n \mathop = n_0}^\infty f_n \left({z}\right) \ne 0$ for $z\in U$.

Let $f$ be identically zero on $U$.

Then $\displaystyle \prod_{n \mathop = 1}^{n_0-1} f_n \left({z}\right)$ is identically zero on $U$.

Then some $f_k$ with $k<n_0$ is identically zero on $U$.

By Uniqueness of Analytic Continuation, $f_k$ is identically zero on $D$.

Now let $f$ be not identically zero on $U$.

We have: