14. Let {A1,A2,A3, . . .} be a sequence of events. Prove that if the series P

∞ n

=1 P

(

An)

converges, then P

T

∞m

=1 S

∞ n

=

m An



= 0. This is called the Borel–Cantelli lemma.

It says that if P

∞ n

=1 P

(

An)

<

∞, the probability that infinitely many of the An’s occur is 0.

Hint: Let Bm =

S

∞ n

=

m An and apply Theorem 1.8 to {

Bm,m

≥

1

}.