Borel-Cantelli Lemma

Let $\{A_n\}_{n=0}^\infty$ be a Sequence of events occurring with a certain probability distribution, and let $A$ be the event consisting of the occurrence of a finite number of events $A_n$, $n=1$, .... Then if

$$\sum_{n=1}^\infty P(A_n)<\infty,$$

then

$$P(A)=1.$$

References

Hazewinkel, M. (Managing Ed.). Encyclopaedia of Mathematics: An Updated and Annotated Translation of the Soviet ``Mathematical Encyclopaedia.'' Dordrecht, Netherlands: Reidel, pp. 435-436, 1988.