Could anyone help me solve the following problem ?

Show that a covering map is proper if and only if it is finite-sheeted.

This is the problem 11-10 from "Introduction to Topological Manifolds" by John M. Lee. The definition of a covering map in this book is the following.

A covering map is a continuous surjective map such that is connected and locally path-connected, and every point of has an evenly covered

neighborhood.

My first attempt was to show that the image of any sequence in that diverges to infinity diverges to infinity in . But, I noticed that for this to be a sufficient condition for properness of , should be a second countable Hausdorff space, and that no additional condition is assumed for other than connectedness and local path-connectedness.

I don't have a clue any more.