Reverse Borel Cantelli

Question

My question is regarding the Borel Cantelli Lemmas, what can we say about their reverse statements?

$X_{i}$ independent in what follows.

For example say $P(X_{n}>n$ i.o$)=0 $ can we say $\sum P(X_{n}\geq n) < \infty$ ? by some reverse BC.2 lemma?