Let be a sequence of distinct points in a domain that accumulates on , and let be a nonempty closed subset of the extended complex plane . Show that there is an analytic function on such that is the set of cluster values of along the sequence .
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 . Thanks in advance.