Let $U$ be an open subset of $\mathbb{C}$ and $\{f_n\}$ be a sequence of holomorphic functions. Suppose that $f_n$ converges uniformly to a function $f$ on compact subsets of $U$ and that $f$ is not identically zero in $U$ and $f(w)=0$ for some $w \in U$.
Prove that there exists $N \in \mathbb{N}$ and a sequence $\{z_n\}$ such that $f_n(z_n) = 0$ for all $n \geq N$ and $\lim_{n \rightarrow \infty}z_n = w$.
How can I prove the existence of $z_n$ and $f_n(z_n)$ is exactly 0? Any idea?
Hint: Let $\Gamma$ be a circle around $w$ such that $\Gamma$ and its interior are in $U$, and $f_n$ is nonzero on $\Gamma$. Then $$\dfrac{1}{2\pi i} \oint_\Gamma \dfrac{f_n'(z)\; dz}{f_n(z)}$$ is the number of zeros of $f_n$ (counted by multiplicity) inside $\Gamma$