Proof(I didn't do it in class):
Uses the notion of moment generating function(MGF).
A way of proof can be seen through the fact (that I didn't prove) that if the generating functions of a sequence of random variables converges to the limiting generating function of a random variable Z then the distribution functions converge to the distribution function of Z.
Reminder: The generating function of the standard Normal is
We show through use of the fact that the moment generating function
of a sum of independent rv's is the product
of their mgf's and the fact that
they have the same distribution
We then prove that the log of this tended to
The theorem is also true for independent variables who do not have the same distributions, along as they are bounded and the means and variances are finite.