Editing Borel–Cantelli lemma (section)

===Example===

Suppose (''X''<sub>''n''</sub>) is a sequence of [[random variable]]s with Pr(''X''<sub>''n''</sub> = 0) = 1/''n''<sup>2</sup> for each ''n''. The probability that ''X''<sub>''n''</sub> = 0 occurs for infinitely many ''n'' is equivalent to the probability of the intersection of infinitely many [''X''<sub>''n''</sub> = 0] events. The intersection of infinitely many such events is a set of outcomes common to all of them. However, the sum ΣPr(''X''<sub>''n''</sub> = 0) converges to {{nowrap|{{pi}}<sup>2</sup>/6 ≈ 1.645 < ∞,}} and so the Borel–Cantelli Lemma states that the set of outcomes that are common to infinitely many such events occurs with probability zero. Hence, the probability of ''X''<sub>''n''</sub> = 0 occurring for infinitely many ''n'' is&nbsp;0. [[Almost surely]] (i.e., with probability 1), ''X''<sub>''n''</sub> is nonzero for all but finitely many&nbsp;''n''.