Let ${X_n}$ with parameter $\lambda$ be a Poisson variable, show that $\frac{1}{n} \sum_{k=1}^{n} X_{n}^{2} \rightarrow \lambda + \lambda^2$ q.c

Question

I can't see how it goes to $\lambda$.

Let ${X}_{n \in \mathbb{N}}$ independent Poisson with parameter $\lambda$ be a Poisson variable, show that $\frac{1}{n} \sum_{k=1}^{n} X_{k}^{2} \rightarrow \lambda + \lambda^2$ q.c

Answer 1

I may be really off, and in case please someone correct me, but I'm inclined to think that the result is wrong.

I'd say that $$\frac{1}{n} \sum_k X_{k}^{2} \to \mathbb{E}[X_1^2] = \lambda + \lambda^2$$