locally compact and proper functions

Hello,

i try to solve following exercise:

Let f:X-> Y be an continuous immersion. X a Hausdorff-space and Y locally compact, i.e. Y is a Hausdorff-space with the property that every point y $\displaystyle \in$ Y has a nbh. U with closure(U) compact.

Show that f is proper iff f^(-1) (K) $\displaystyle \subset$ X is compact for all K $\displaystyle \subset$ Y compact.

"=>" This direction was not so difficult. I could solve it.

But the other direction "<=" i couldn't solve.

We have shown before, that "f is proper" is equivalent to "f is closed" and to "f is an embedding".

I try to show that f is closed, since i think this must be the most easy way.

Have you some help for me.

Thanks a lot