sequence of cumulative distribution functions

Xk - sequence of independent random variables, Fk - cumulative distr functions, that are continuous and strictly increasing.

Zn = (1/n^(1/2)) SUM( 1 + log (1-Fk)) n=1,2, ...
Show that for n->inf Zn converges to a normal distribution with mean 0 and variance 1.

I guess the centrl limit theorem should be used here. But I don't understand what to do with these log of probability distribution function.
Thanks for your help.