# How to prove limit of an infinite sum

I have problems with this probability exercise

Let $$f : \mathbb{R} \to \mathbb{R}$$ a continuous and bounded function. Prove that for every $$\lambda > 0$$

$$\lim_{n \to \infty}\sum_{k=1}^{\infty} f \left(\frac{k}{n} \right)\frac{(n\lambda)^k}{k!}e^{-n\lambda}=f(\lambda)$$

I think I could prove this with epsilon-delta definition. Then, I tried the next but I am not sure.

\begin{align*} \left|\sum_{k=1}^{\infty} f \left(\frac{k}{n} \right)\frac{(n\lambda)^k}{k!}e^{-n\lambda}-f(\lambda)\right| &= \left|\sum_{k=1}^{\infty} f \left(\frac{k}{n} \right)\frac{(n\lambda)^k}{k!}e^{-n\lambda}-f(\lambda)\sum_{k=1}^{\infty}\frac{(n\lambda)^k}{k!}e^{-n\lambda}\right|\\ &=\left|\sum_{k=1}^{\infty} \left(f \left(\frac{k}{n} \right)-f(\lambda)\right) \frac{(n\lambda)^k}{k!}e^{-n\lambda}\right|\\ \end{align*}

I guess the next step is to use the fact that function is continuous and bounded but I can argue exactly why. Any help?

First, observe that $$\mathbb{E}\left[f\left(\frac{N_{n\lambda}}{n}\right)\right] = \sum_{k = 0}^\infty f\left(\frac{k}{n}\right)\frac{(n\lambda)^k}{k!}e^{-n\lambda} = \sum_{k = 1}^\infty f\left(\frac{k}{n}\right)\frac{(n\lambda)^k}{k!}e^{-n\lambda} + e^{-n \lambda},$$ where $$N_{n\lambda}$$ is a Poisson random variable with parameter $$n\lambda$$. Thus the quantity inside the limit is $$\mathbb{E}\left[f\left(\frac{N_{n\lambda}}{n}\right)\right] - e^{-n\lambda}$$.
It is a well know fact about the sum of Poisson random variables that if $$P_1, \dots, P_N$$ are independent Poisson random variables with parameters $$\lambda_1, \dots, \lambda_N$$, then their sum is also Poisson with parameter $$\sum_{i =1}^N \lambda_i$$. Therefore, $$N_{n\lambda} \overset{law}{=} \sum_{i =1}^N N_{\lambda}^i$$ with the $$N_{\lambda}^i$$'s being independent. As a consequence we have, by the strong law of large numbers,
$$\frac{N_{n\lambda}}{n} \to \mathbb{E}\left[N_\lambda\right] = \lambda \quad \text{almost surely}.$$ Then by dominated convergence we have that since $$f$$ is bounded $$\lim_{n\to \infty}\mathbb{E}\left[f\left(\frac{N_{n\lambda}}{n}\right)\right] = f\left(\lambda\right).$$
Injecting this result into the limit we want to solve gives that $$\lim_{n \to \infty}\sum_{k=1}^{\infty} f \left(\frac{k}{n} \right)\frac{(n\lambda)^k}{k!}e^{-n\lambda} = \lim_{n\to \infty} \left[\mathbb{E}\left[f\left(\frac{N_{n\lambda}}{n}\right)\right] - e^{-n\lambda}\right] = f\left(\lambda\right).$$