(BorelCantelli lemma). Let {An} be a sequence of events in a sample space , such that the sequence of real numbers converges to a real

(Borel–Cantelli lemma). Let {An} be a sequence of events in a sample space Ω, such that the sequence of real numbers

converges to a real number. Prove that

that is, the probability that infinitely many, among the events A1, A2,…, occur is zero.
(Hint: Apply Proposition 1.11 to the sequence of events {Br}r≥1, where Br = ∪∞ n=r An for r = 1, 2,….)

a = P(A), ), 1 = 1,2,..... i=1