Example of a connected Riemannian manifold containing a non-compact closed bounded set

Question

What is an example of a connected Riemannian manifold containing a non-compact closed bounded set?

By the Hopf-Rinow Theorem, I know that the closed bounded sets of a connected Riemannian manifold are compact if and only if the manifold is complete. So I have to pick a non complete Riemannian manifold. Let's say I choose the punctured sphere or an open hemisphere with the induced metric from Euclidean space. "Where is" that closed bounded subset of the sphere which is not compact? Or am I in the wrong track?

Answer 1

In any metric space, if $(x_n)$ is a Cauchy sequence which does not converge, then the subset $\{x_n\}$ is closed, bounded, and not compact. Alternatively, if your entire manifold is bounded (as in the examples you name), you can just take the closed bounded set to be the entire manifold.

Answer 2

Take the open unit disk in the plane. It is closed in itself but not compact.