$f_n \to f$ uniformly on compact subsets of $D$ ; $f$ non-constant , then there is a sequence $\{z_n\}$ s.t. $f_n(z_n)=f(z) , \forall n >N$

Question

Let $D$ be an open connected set in $\mathbb C$ and $\{f_n \}$ be a sequence of holomorphic functions in $D$ such that $f_n \to f$ uniformly on compact subsets of $D$ . If $f$ is non-constant and $z \in D$ then how to show that there exist a sequence $\{z_n\}$ in $D$ and positive integer $N$ such that $f_n(z_n)=f(z) , \forall n >N$ ? I think I have to apply Hurwitz's theorem , but I can't really see how . Please help . Thanks in advance

Answer 1

hint: Assume for contradiction that the stated is not true. Consider the sequence of functions $w\mapsto g_n(w):=f_n(w)-f(z)$. Relate this hint to the one you gave yourself.