Let $f: \mathbb R^n \to \mathbb R^m$, $(1 \leq n \leq m)$ be a injective Lipschitz function, it is true that $f^{-1}(K)$ is compact for all compact $K \subset \mathbb R^m$, i.e $f$ is a proper map.
I'm interested to know if that fact is true, I try to found some counterexamples but I can't found it, any hint or suggestions I will be very grateful
Take for example $\text{arctan}:\mathbb{R}\to\mathbb{R}$. Clearly the preimage of $[-\pi/2,\pi/2]$ is $\mathbb{R}$. The statement is more interesting if you restrict the codomain to the image of $f$.