Question: Let $\Omega \subset \mathbb{C}$ be open and connected. Suppose $f_n: \Omega \to \mathbb{C}$, $n \ge 1$, $f: \Omega \to \mathbb{C}$ holomorphic s.t. $f_n \to f $uniformly on each ball $B \to \Omega$, with $\overline B \subset \Omega$. Suppose that $f_n$ has no zeros. Show that either $f = 0$ or $f$ has no zeros in $\Omega$.
Attempt:
Assume that $f \ne 0$. Then, assume that there exists $a \in \mathbb{C}$ such that $f(a) = 0$ for some isolated $a \in \Omega$. Then, we can find a $B_r(a)$ such that $f$ is never zero, $r \gt 0$, small enough.
Then, for any $\epsilon \gt 0$, there exists $N \in \mathbb{N}$, such that given a ball around a, $\left|f(z) - f_n(z)\right| \lt \epsilon$, for $n \ge N$.
After that, I can not find where to apply Rouche. Maybe we can do $\left|\left|f(z)\right| - \left|f_n(z)\right|\right| \lt \left|f(z) - f_n(z)\right| \lt \epsilon$, in somewhere to find whether $\left| f \right| > \left| f_n(z) \right|$ or viceversa.