Let $f$ be a holomorphic function on an open set $G$, such that $f$ has a finite number of zeros, then we can write $f$ as $f = p*g$, where $p$ is a Polynom and $g$ a holomorphic function on $G$ without zeros.
I´ve tried using the contradiction and Liouville´s theorem, but not sure if that is the right way.
Edit:
So my new solution is this but I´m still not sure if it´s right
Let $(z_{1},z_{2},z_{3},...,z_{n}), n\in\mathbb{N}$ be the set of zeros of $f$, then we define: $$ p(z) =\prod_{j=1:n}(z-z_{n})$$ which obviously has the same zeros as $f$. Then $f/p$ has no zeros, because they both have zeros at the same points. Then we can define $$g : = f/p$$ and $g$ ish holomophic, because $f$ and $g$ are also.