Let $M$ and $N$ be smooth, connected $n$-dimensional manifolds. Let $M$ be compact and non-empty.
Show that every embedding $f: M \to N$ is a diffeomorphism.
So because $f$ is a embedding we have that $f(M) \subset N$ is a submanifold and especially that $f: M \to f(M)$ is a diffeomorphism. My idea would be to show that $f(M) = N$ and that's somehow related to $M$ being compact but I don't really know how.