[SOLVED] Complex limit points

This is the first of a probably numerous series (pun intended) of questions about complex series. My series work has never been good, and I don't write proofs well. (Headbang)

So here we go with the first:

Quote:

A point $\displaystyle \alpha$ is said to be a limit point of a set E of points in the complex plane (as opposed to a sequence $\displaystyle z_n$) if every neighborhood of $\displaystyle \alpha$ contains infinitely many (distinct) points of E. Show that a sequence $\displaystyle z_n$ can have a limit point $\displaystyle \alpha$ in the sense that, given any $\displaystyle \epsilon$ the inequality $\displaystyle |z_n - \alpha | < \epsilon$ holds for infinitely many points of $\displaystyle z_n$, without being a limit point of the set of points corresponding to the terms of $\displaystyle z_n$.

I can't even think of an example that fits this situation, much less writing a proof for it. Any pointers (not full solutions please) would be appreciated.

-Dan