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.
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.