As I recall in this version the random variables are independent with finite variances but the variance need not all be the same.  The CLT result holds under a somewhat complicated condition called the Lindeberg condition and the traditional proofs use transform methods.  
But the proof we learned was probabilistic.  It involved splitting the sum into two pieces.  One piece converged to N(0,1) in distribution and the other converge to 0 in probability.  This technique was used because it was much easier to show the first sum satisfied the CLT.  But the fact that the second sum was negligible was harder.  The following link gives an interesting paper by Larry Goldstein that give a probabilistic proof of the Linderberg Feller Theorem that is very similar or the same.  It also may be of interest to the OP because it includes some history on the CLT.
http://bcf.usc.edu/~larry/papers/pdf/lin.pdf