almost surely convergence of poisson variables

Question

I'm having trouble with this question, would love to get help!

Let $\{X_n\}$ be a series of variables where $X_n \sim \text{Pois}(\lambda n)$. Show that if $X_n\to 0$ almost surely, then $\lambda n \to 0$.

Thank you:)

Answer 1

If a sequence of R.V converges almost surely then it converges in probability. Thus $$\mathbb{P}(X_n = 0) \to 1.$$ In other words $$e^{-\lambda_n} \to 1$$ so $$\lambda_n \to 0.$$