2.40 Use the Borel–Cantelli lemma (Lemma 2.5) to prove the following:

(i) If for every  > 0 we have

∞

n=1 P(|ξn −ξ| ≥ ) < ∞, then ξn

−a→.s. ξ as n→∞.

(ii) If

∞

n=1 E(|ξn − ξ |p) < ∞for somep > 0, then ξn

−a→.s. ξ as n→∞.