Is a finite Hausdorff space necessarily discrete?
Just thought of another answer to this. Any topology on a finite set is compact. Any map from a discrete topology is continuous. Hence by the famous theorem on maps from compact spaces into Hausdorff spaces, the identity map on a finite space is a homeomorphism from the discrete topology to the given Hausdoff topology.
A certain phrase involving sledgehammers and walnuts springs to mind, though.