Suppose Xn is a sequence of real-valued random variables.

(i) Assume Xn is Cauchy in probability; that is, for all  > 0, lim min(m,n)→∞ P{|Xn − Xm| > } → 0 .

Then, show there exists a random variable X such that Xn P

→ X, in which case we may write X = limn→∞ Xn.

(ii) Assume Xn satisfies E(|Xn|p) < ∞. Also, assume Xn is Cauchy in L p; that is, lim min(m,n)→∞ E(|Xn − Xm|

p) → 0 .

Then, show there exist a random variable X such that E(|Xn − X|p) → 0 and E(|X|p) < ∞.