I am looking for a proof of the CLT in some simple case that does not rely on moment generating functions or characteristic functions. I think I can see how to do this in the case of Bernoulli summands, but I would be very interested to know if there is a bare hands proof out there somewhere that works for (for example) random variables taking finitely many values.
What I'm hoping for is either a reference to such a proof; or the outline of such a proof if anyone knows one. Thanks for any pointers!