Injection from compact to Hausdorff implies embedding

Statement
Any injective continuous map from a compact space to a Hausdorff space is an embedding; in other words, it is a homeomorphism to its image, when the image is given the subspace topology.

Proof idea
We use two facts:


 * any continuous map from a compact space to a Hausdorff space is closed
 * any closed injective map is a subspace embedding