=+5. Consider a sequence of independent events A1, A2,... satisfying

∞

i=1 Pr(Ai) = ∞.

As a partial converse to the Borel-Cantelli lemma, prove that infinitely many of the Ai occur. (Hints: Express the event that infinitely many of the events occur as ∩∞

n=1 ∪∞

i=n Ai. Use the inequality 1 − x ≤ e−x to bound an infinite product by the exponential of an infinite sum.)

1.9 Problems 21