Let $f:R^2\to R^2$ be a continuously differentiable function such that $Df(x)$ is invertible for all $x\in R^2$ and $f^{-1}(K)$ is compact for every compact set $K$. Show that $f$ is surjective.
The first condition gives by the inverse function theorem that $f$ is invertible in a neighborhood of any point. But how exactly to use properness?
The fact that $Df$ is invertible implies that the image of $f$ is open. It is enough to show that the image of $f$ is closed.
Suppose that $y=lim_nf(x_n)$, let $B=B(y,1)$, the closed ball of radius $1$, there exists $N$ such that $n>N$ implies that $f(x_n)\in B$, $f^{-1}(B)$ is compact and contains $x_n, n>N$, since $f^{-1}(B)$ compact, you can extract a converging subsequence $x_{n_p}$, write $x=lim_nx_{n_p}$ since $f$ is continuous, $f(x)=lim_nf(x_{n_p})=y$.