Let S be an infinite subset of $\displaystyle \overline{\mathbb{C}}$ (the extended complex plane). Show that either S has a limit point in S or every Disc D'($\displaystyle \infty$,r) (punctured disc) contains a point of S (so that $\displaystyle \infty$ is a limit point of S in the space $\displaystyle \mathbb{C}$).

My argument so far is:

Assume S does not have a limit point in S. Then take an abitrary disc D'($\displaystyle \infty$,r).

Now assume for contradiction D'($\displaystyle \infty$,r) does not contain a point of S. But I can't see where I can find a contradiction to the first statement. Can someone help, or provide an alternative argument? Thanks