Subgroups and normal subgroups

Hey I have a question out of my algebra text (Hungerford, p.92 #13)

It asks to show that if G is a group containing a proper subgroup H of finite index, then it contains a proper normal subgroup of finite index.

It's easy to show the fact if G is finite or abelian, but I'm left with the last case if G is nonabelian and infinite. I'm trying to work with the center, which I know is normal, but I don't know how to manipulate it to create a resultant normal group of finite index, and obviously I need to incorporate the given subgroup H somehow.

Letting C be the center of G, I tried to consider the group HC = { hc | h in H, c in C} but I get no success trying to show this is either normal or has finite index. I am terrible with conjugation in groups in general, and considering this exercise involves a degree of counting cosets, I'm suspecting conjugation is going to play a part, which is even more intimidating.

Any assistance would be appreciated.