[30] In an infinite sequence of outcomes generated by a (p, 1−p)

Bernoulli process, let A1, A2,... be an infinite sequence of events each of which depends only on a finite number of trials in the sequence. Denote the probability of Ak occurring by Pk. (Ak may be the event that k consecutive 1s occur between the 2kth trial and the 2k+1th trial. Then Pk ≤ (2p)k.)

(a) Prove that if Pk converges, then with probability one only finitely many Ak occur.

(b) Prove that if the events Ak are mutually independent, and if Pk diverges, then with probability one infinitely many Ak occur.

Comments. These two assertions are known as the Borel–Cantelli Lemmas. Source: [W. Feller, Ibid.].