Let $U$ be an open connected subset of $\mathbb{C}^n$, and $O(U)$ the ring of holomorphic functions on $U$. Prove that $O(U)$ is an integral domain.
I have done
If $fg\equiv0$ in $U$, then $f$ vanishes at a finite or infinite number of points of $ U $
If $f$ vanishes in $A$, $A$ finite, then $g$ vanishe in $U-A$, $U-A$ is nonempty open content in $ U $, then $g\equiv0$ in $U$
Assume $f$ is not identically $0$. Then there is a point $p$ with $f(p) \neq 0$. By continuity of $f$, there is a neighborhood $U$ of $p$ with $f(z) \neq 0$ for $z \in U$. Can you take the argument from here?