Borel-Cantelli Lemmata to Kochen-Stone Theorem

Lemmata to Kochen-Stone Theorem

Lemma $10$

Let $A_n$ be a sequence of events with $\ds \sum \map \Pr {A_n} = \infty$ and:

$\ds \liminf_{k \mathop \to \infty} \frac {\ds \sum_{1 \mathop \le m, n \mathop \le k} \map \Pr {A_n \cap A_m} } {\ds \paren {\sum_{n \mathop = 1}^k \map \Pr {A_n} }^2} < \infty$

Then there is a positive probability that $A_n$ occur infinitely often.