The Borel-Cantelli Lemma asserts that let (X,Σ,μ) be a measure Space, if (An) is a sequence of measurable sets such that





Then 



Proof :

If A1,A2,...,An are sequences, the Borel-Cantelli Lemma States that if the sum of the measure of the An is finite then the measure that infinitely many of them occur is zero(0) that is 





By definition of Limit Supremum:







Now we can use the concept of monotonicity and intersection of subsets 





Now by countable subadditivity we have 







However by assumption we have






Converges. And by convergence series, the series tends to zero. 





By the properties of lower and upper bounds of sequence, we have 








But as μ is a measure, the converse inequality also holds. Hence