Given a continuous function $g$ I am trying to show that $\sum_{k=0}^{\infty}g(\frac{k}{n})\frac{(\lambda n)^ke^{-\lambda n}}{k!} \overset{D}{\to}g(\lambda)$.
I think the idea is probably to use the continuous mapping theorem. If we define $X_n \sim P(\lambda n)$, then the summation can be rewritten like $\sum_{k=0}^{\infty}g(\frac{k}{n})P(X_n = k)$. Now I would like to apply the continuous mapping theorem for $E[g(X_n)]$ somehow. However, $E[g(X_n)] = \sum_{k=0}^{\infty}g(k)\frac{(\lambda n)^ke^{-\lambda n}}{k!}$ which is close but not exactly the same to the original term. Any suggestions?