Question 8-8 in Lee's Introduction to smooth manifolds asks us to show that if $M \subset N$ is an embedded submanifold then it is closed iff the inclusion map is proper. Equivalently, a smooth embedding $g:M \to N$ is proper iff its image is closed.
Do we really need embedding for this -or even the smoothness? It seems to me, that the only relevant hypothesis is that the topologies on $M,N$ are metrizable, and that $g$ is a homeomorphism onto its image.
Any ideas?