Hopf-Rinow Theorem tell us: $M$ a Riemannian manifold and every closed and bounded subset of $M$ is compact . Then $M$ is complete.

Question

Hopf-Rinow tells us :$M$ is a Riemannian manifold such that every closed and bounded subset of $M$ is compact , then $M$ is complete.

Many lectures say this can be proved by the general topology. But this puzzles me.

We need to show that every Cauchy sequence converge to some point in $M$.

idea:$\{x_n\}$ is a cauchy sequence and thus bounded.

1. How could we say $\overline{ \{x_n\}}$ is bounded ,too.

2. $\overline{ \{x_n\}}$ is bounded , then it is compact by hypothesis . How to choose a subesquence of $\{x_n\}$ s.t. it converges, we only know that $\overline{ \{x_n\}}$ is compact and $\{x_n\}$ may not be compact.