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

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 $\displaystyle f(z)$. Thanks in advance.