Suppose $f :X \to Y$ is a continuous proper map between locally compact Hausdorff spaces. Show that if $f$ is a surjective map which is locally injective then $f$ must be a covering map.
It is well known that $f$ is a closed map. Suppose $y \in Y$ be any element then by properness $f^{-1}(y)$ must be a finite set. Suppose $f^{-1}(y)$ ={$x_1,x_2,...,x_n$} then as $X$ is Hausdorff we get open nbhds $U_i$'s separating $x_i$'s. But I am unable to see how to proceed now. Thank you for your help.