Let $X,\ Y$ be manifolds in $\mathbb R^n$ (they are locally diffeomorphic to open subsets of some euclidean space.) I have a question about this fact (an exercise in Guillemin and Pollack):
An injective local diffeomorphism $f: X\rightarrow Y$ is a diffeomorphism onto an open subset of $Y$.
This seems too trivial to me and hence I think I musunderstand something. I would prove this claim as follows.
The map $f: X\rightarrow f(X)$ is bijective. It is differentiable at any point since it is locally smooth (and even locally diffeomorphic), and the inverse $f^{-1}: f(X)\rightarrow X$ is also differentiable at any point because given any point $f(x)\in f(X)$, the map $f$ is a diffeomorphism near $x$, so $f^{-1}$ is differentiable at $f(x)$. Thus $f$ is a global diffeomorphism. Because of this and since $X$ is open in itself, $f(X)$ is open in $Y$.
How bad/good does this proof look?