TITLE: Global invertibility of a map $\mathbb{R}^n\to \mathbb{R}^n$ from everywhere local invertibility
QUESTION [39 upvotes]: I was told by a tutor that if $f: \mathbb{R}^n \longrightarrow \mathbb{R}^n$ has an invertible Jacobian Matrix for all $x \in \mathbb{R}^n$ and $\lim_{|x_k| \rightarrow \infty}|f(x_k)|=\infty$ for all such sequences, then $f$ is already globally bijective.
This was very surprising for me (as this seems to be a very strong statement), so I tried to search on the internet for this theorem but I only came up with theorems about local inverses. I hope someone here can give me a reference or a name of this theorem so I could read a detailed proof. I sanity checked it for $n=1$ where it works perfectly.

REPLY [10 votes]: Because $f$ is proper and locally diffeomorphic, $f:\mathbb R^n\to \mathbb R^n$ is an universal covering map. Since $\mathbb R^n$ is simply-connected, the deck transformation group is trivial and therefore $f$ is injective.
The same method can be applied to the general theorem of Hadamard.