Now, let’s finish the proof of the Hopf-Rinow theorem (the first one) started yesterday. We need to show that given a Riemannian manifold which is a metric space , the existence of arbitrary geodesics from implies that is complete with respect to . Actually, this is slightly stronger than what H-R states: geodesic completeness at one point implies completeness.

The first thing to notice is that is smooth by the global smoothness theorem and the assumption that arbitrary geodesics from exist. Moreover, it is surjective by the second Hopf-Rinow theorem.

Now fix a -Cauchy sequence . We will show that it converges. Draw minimal geodesics travelling at unit speed with