I have been looking for a proof of the fact that for a large parameter lambda, the Poisson distribution tends to a Normal distribution. I know the classic proof using the Central Limit Theorem, but I need a simpler one using just limits and the corresponding probability density functions. I was told this was really easy using Stirling's approximation: n! ~ sqrt(2*pi*n) * (n/e)^n but I just don't see it. Anyone knows this proof?