As is well known, the $\mathsf{Binomial}(n,p)$ distribution converges to the $\mathsf{Poisson}(a)$ distribution as $n\rightarrow \infty$, $p\rightarrow 0$ with $np=a$.
I'm pretty sure that the moments of $\mathsf{Binomial}(n,p)$ also converge to those of $\mathsf{Poisson}(a)$, but I don't know how to prove it. Convergence in distribution doesn't imply convergence of moments, in general. How can I prove that the moments converge?
I've found that the binomial probability (mass) function converges uniformly to the Poisson one. This is stronger than convergence in distribution, so perhaps it can be exploited (but if so I don't know how).