Suppose that $f_{n}$ is a sequence in $H(G)$, $f$ is a non-constant function and $f_{n}\to f$ in $H(G)$. Let $a\in G$ and $f(a)=\alpha$. Show that there is a sequence $a_{n}$ in $G$, such that (i)$a=lim_{n \to \infty} a_{n}$ (ii) $f_{n}(a_{n})=\alpha$ for sufficiently large $n$.
I am stuck with this problem and I have no idea. It seem it is an application of Hurwitz's theorem but I am not able to apply it!
Thanks in advance!
Let $\Gamma$ be a positively oriented circle around $a$ such that $\Gamma$ and its interior are contained in $G$. For any $g \in H(G)$ such that $g \ne \alpha$ on $\Gamma$, by the Residue Theorem
$$ M(g) = \dfrac{1}{2\pi i} \oint_\Gamma \dfrac{ g'(z)\; dz}{g(z) - \alpha}$$ is the number of roots of $g - \alpha$ inside $\Gamma$, counted by multiplicity. In particular, if $g = f_n$ is sufficiently close to $f$ on $\Gamma$ this will be at least $1$. You can simply define $a_n$ to be the closest root of $f_n - \alpha$ to $a$, breaking ties arbitrarily.