Riemannian Manifolds and Completeness

[Kreizhn]

Homework Statement

Suppose that for every smooth Riemannian metric on a manifold M, M is complete. Show that M is compact.

2. The attempt at a solution

I'm honestly not too sure how to start this question. If we could show that the manifold is totally bounded we would be done, but I'm not sure how to get that out of the assumption. Any ideas that I could play around with?

 

[Kreizhn]

So I showed that M must be a closed manifold. By Hopf-Rinow if I can show it's bounded then I'll be done. I didn't use the invariance of completeness under arbitrary metrics in my closed argument so I think it will come in use for the bounded part.