Borel–Cantelli lemma

Definition:
Let a sequence  of random variables $(X_n)$:

1. If $\sum_{n  = 1}^{\infty}  Pr(X_n) < \infty$ then $$ Pr(\limsup_{n \rightarrow \infty}  X_n) = Pr(X_n  infinitly  often) = 0$$
2. If  $\sum_{n  = 1}^{\infty}  Pr(X_n) = \infty$ and the events $X_n$ are independent then $$ Pr(\limsup_{n \rightarrow \infty}  X_n) = Pr(X_n  infinitly  often) = 1$$