Image of smooth manifold is a submanifold

Question

It's know that if $M$ is a compact, smooth manifold of dimension $n$ and the map $f: M \to \mathbb{R^m}$ is injective, smooth, $n \le m$ and $Jf(a)$, the Jacobian, has rank $n$ for every $a \in M$, then $f(M)$ is a submanifold of $\mathbb{R^m}$. I'm trying to think of a counterexample to this statement if we suppose $M$ is not compact, but haven't gotten anywhere. Can anyone offer a hint?