Borel-Cantelli Lemma for Poisson random variables

Question

I want to solve a simple problem:

I have a sequence of independent random variables $X_n$ distributed according to Poisson with expectation $\mathbb{E}_{X_n} = 1$

I want to prove that $\mathbb{P}\{\lim \sup_{n\to\infty} (\frac{X_n \cdot \ln \ln n}{\ln n}=1)\}=1$

I see that I need to use Borel-Cantelli Lemma. I think all I need to do is to prove that $\forall M \sum_{n=1}^{\infty} \mathbb{P}(\frac{X_n \cdot \ln \ln n}{\ln n}=M)$ diverges. But I definitely don't understand how to show that.

Answer 1

This is false. Poisson random variables take only non negative integer values. So $P(\frac {X_n n \ln \ln n} {\ln n}=1)=0$ for every $n$ which implies that $\lim \sup$ of these events also has probability $0$.