The Borel-Cantelli lemma guarantees with probability 1 that only finitely many events in a sequence
occur when the sum of their probabilities converges. Let 
be a sequence
of events occurring with a certain probability distribution, and let 
be the event consisting of the occurrence of a finite number
of events 
for 
,
2, .... Then the probability of an infinite number of the 
occurring is zero if
 |
Equivalently, in the extreme case of 
for all 
, the probability that none of them occurs is 1 and, in particular,
the probability of 
that a finite number occur is also 1.
