Three proofs of a limit theorem

We show three diferent proofs of the central limit theorem using elementary methods. The central limit theorem with the Feller - Lindeberg condition is proven using a convergence of charakteristic functions and Fejer theorem about uniform convergence of trigonometric polynoms on a bounded interval. The second proof is based on the fact that convergence in distribution is equivalent to convergence of means of functions with all derivatives bounded. The central limit theorem for sums of independent random variables with all moments finite is shown using convergence of all moments and determinacy of normal distribution by its moments