Let (an) be a convergent sequence. Show that the set S = {an : n in N, the natural numbers} (the range of the sequence (an)) has a maximal or a minimal element.

If I assume that the sequence has no maximal or minimal element, then I know that both the inf(an) and the sup(an) are not in the set S. I don't see where the contradiction happens though. Could someone help me here?