Complex Analysis and Limit point help So S is a complex sequence (an from n=1 to infinity) has limit points which form a set E of limit points. How do I prove that every limit point of E are also members of the set E. I think epsilons will need to be used but I'm not sure.
Thanks.

Let $z$ be a limit point of $E$, and take any $\varepsilon>0$. There is some $x\in E$ with $\lvert x-z\rvert<\varepsilon/2$. And since $x\in E$, there are infinitely many members of $S$ within an $\varepsilon/2$-ball around $x$. They will all be within an $\varepsilon$-ball around $z$, and you're done.
