Let $\{X_{n}\}_{n}$ be a sequence of random variables independent and iddentically distributed with distribution $P$ and defined in the same probability space $(\Omega,\sigma,\mathbb{P})$. Let $A$ be a Borel-set such that $P(A) > 0$. We want to show that $$ \mathbb{P}(\{\omega\in\Omega : \#(\{n\in\mathbb{N} : X_{n}(\omega)\in A\}) = \infty\}) = 1 $$
I'm stuck in this result. I think that the proof is based in the fact that the set $A$ has positive measure, but I'm not able to related with the probability.
This an immediate consequnce of Borel-Cantelli Lemma: $\sum P(X_n \in A)=\sum P(X_1 \in A)=\infty$ and hence the result. Ref: https://en.wikipedia.org/wiki/Borel%E2%80%93Cantelli_lemma