If $f$ is an entire function, there exists an unbounded sequence $z_n$ such that $f(z_n)$ converges

Question

Let $f$ be an entire function which is not a polynomial. Then for every $c\in\mathbb{C}$, there exists an unbounded sequence $(z_n)_{n\in\mathbb{N}}$ with $f(z_n)\to c$ if $n\to \infty$.

I am thinking of somehow using Liousville's theorem, but I don't see how to invoke it.