Borel-Cantelli Lemmas

In the setting of Probability Spaces: Notation, let $(A_{k})_{k\in\mathbb{N}}$ be a sequence of events.

(i) First Borel–Cantelli lemma. If $\sum_{k\in\mathbb{N}}P(A_{k})<\infty$, the sum taken in $[0,\infty]$, then

$$P(\{A_{k}\text{ infinitely often}\})=0.$$

(ii) Second Borel–Cantelli lemma. If the events $A_{k}$, $k\in\mathbb{N}$, are independent and $\sum_{k\in\mathbb{N}}P(A_{k})=\infty$, then

$$P(\{A_{k}\text{ infinitely often}\})=1.$$