Normally convergent products define holomorphic functions with the expected zeros

Fix a compact set $K\subseteq \mathrm{\Omega }$. By [F1], choose $N$ so that $f_n$ has no zero on $K$ for $n\ge N$ and $M_n:={\sup }_K\mathrm{\mid }{f}_n-1\mathrm{\mid }$ satisfies ${\sum }_{n\ge N}{M}_n<\mathrm{\infty }$; enlarging $N$ if needed, assume also $M_n<1\mathrm{/}2$ for $n\ge N$.

For $m\ge \mathrm{\ell }\ge N$ and $z\in K$, one has $\mid {\prod }_{n=\mathrm{\ell }}^m{f}_n(z)-1\mid \le {\prod }_{n=\mathrm{\ell }}^m(1+M_n)-1$, while $\mathrm{\mid }{f}_n(z)\mathrm{\mid }\ge 1-M_n>1\mathrm{/}2$ for $n\ge N$. By [F2], the real products ${\prod }_{n\ge N}(1+M_n)$ and ${\prod }_{n\ge N}(1-M_n)$ converge, and the latter has positive value because all its factors are positive. Thus the tail partial products of $\prod f_n$ are uniformly bounded above and uniformly bounded away from $0$ on $K$.

The estimate of step 2.1 implies that the tail partial products are uniformly Cauchy on $K$, hence converge uniformly there to a continuous zero-free limit $Q_K$; multiplying by the finite holomorphic prefix ${\prod }_{n<N}{f}_n$ gives uniform convergence of the full partial products on $K$. Because $K$ was arbitrary, the convergence is locally uniform on $\mathrm{\Omega }$, and [F3] makes the limit function $F$ holomorphic.

On the fixed compact set $K$, write $F=\left({\prod }_{n<N}{f}_n\right)Q_K$ with $Q_K$ holomorphic and zero-free by step 3.1. If no factor vanishes identically on any connected component of $\mathrm{\Omega }$, the finitely many prefix factors have only isolated zeros on their respective components, and [F4] shows that every zero of $F$ on $K$, with its multiplicity, comes from that finite prefix and no tail factor contributes a new zero.