Central Limit Theorem (CLT) is one of the two most important theorems in statistics (the other is the Large Number Theorem). In this note, the theorem of moment generating function is used to prove the CLT. This is a weak proof, because the underlying logic is that by showing two varibles have the same mgf, we jump to conclude that they are also the same on pdf. We do not dig further on the property of moment generating function.
Notice also that moment generating function may not always exist for all functions, so sometimes people use characteristic function instead to prove CLT.
