I am asked to prove that if then and

So I assume I have to treat these as sequence because cannot be a function of z since the n is there. I know the definition of the limit but I just get stuck. In were real I think I would take the log and proceed from there, but that seems considerably harder with the complex logarithm.

Anyway I have:

Let . Now I have to find a such that when , which I cannot do.

Can anyone help me... thanks.