I've been working on the following exercise and got stuck:
"Prove that a connected Riemannian manifold is complete if and only if every regular curve that diverges to infinity has infinite length."
Here, a curve with domain $[0,b)$ (and $b$ possibly infinite) diverges to infinity if for every compact subset of the manifold, there exists a time beyond which it does not intersect the compact set anymore.
I was able to prove the forward direction, but I can't figure out how to get the converse. I've tried both to show that maximal geodesics are defined for all time, and when that didn't work, I tried to show Cauchy sequences converge, but I didn't have success in either case. Any advice is greatly appreciated.