Upward stability of Riemannian completeness ($\tilde{g}\ge g,g\text{ complete}⟹\tilde{g}\text{ complete}$)

Pointwise domination of Riemannian metrics gives domination of their distances. A Cauchy sequence for the larger distance is Cauchy for the complete smaller one. Smooth positive metrics induce the same manifold topology, so its limit is also a limit in the larger distance. The Hopf-Rinow theorem turns metric completeness into geodesic completeness. Mere positivity of the second metric, without domination, is insufficient.