Here are two supposedly equivalent definitions of a smooth isotopy (M and N are smooth manifolds):
A smooth level preserving imbedding $M \times I \rightarrow N \times I$
A smooth map $ F: M\times I \rightarrow N$ such that each $F_t$ is an imbedding
For the life of me I cannot show that an $F$ satisfying 2 will induce an imbedding as in 1. Clearly the map will be an injective immersion but why is it a homeomorphism onto its image? I tried showing its proper but I can't see why a sequence of points escaping from compact sets in M couldn't be mapped into a single compact set in N at different times, even though the times converge to some time in the interval. Please help! I'm probably just being very stupid. Or perhaps there's a counter example?