Show that for any $w \in \mathbb{C}$ there exists a sequence $z_n$ s.t. $f(z_n) \rightarrow w$ I want to prove the following:
Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be holomorphic and non-constant. Then for $w \in \mathbb{C}$ there exists a sequence $(z_n)_{n \in \mathbb{N}} \subset \mathbb{C}$ with $lim_{n\rightarrow w}f(z_n) = w$.
Which theorem can I use here? I know that by Liouville $f$ must be unbounded but does that help me?
Can someone give me a hint?

If the statement is untrue for a $w$, then $\frac{1}{f(z)-w}$ becomes holomorphic -- since the denominator is away from zero by a positive $\epsilon _0 $ distance. But then it is also bounded, which proves that it is constant. This cannot be ok because $f$ should not be constant.
