Closed singleton embedding in a Hausdorff hyperspace

The map $x\mapsto \{x\}$ is an isometry into the hyperspace of nonempty closed bounded subsets. Its image is closed because a Hausdorff limit of singletons has diameter zero and is therefore a singleton.

If the Hausdorff hyperspace is complete, its closed subspace of singletons is complete. Since that subspace is isometric to the original metric space, the original space is complete as well.