Let \{ z_k \} be a sequence of distinct points in a domain D that accumulates on \partial D, and let E be a nonempty closed subset of the extended complex plane \overline{\mathbb{C}}. Show that there is an analytic function f(z) on D such that E is the set of cluster values of f(z) along the sequence \{ z_k \}.

I am not sure how to prove this. I would appreciate a few hints or suggestions. In this section we have covered Runge's Theorem. However, I am still not sure how to prove this. I don't see how to show the existence of such an analytic function f(z). Thanks in advance.