Problem 3. Let {Xn}nen be a sequence of random variables, and let X be another random variable. Assume that for all e > 0, ~ P(X) - X2 ) < . п n=1 Prove that Xn converges almost surely to X. Hint: Define the event : = An := {w€2:3€ >0 such that Xn(w) - X(w) > €}, and use it to show that if for each € > 0 we have P({]Xn – X| > e} i.o.) = 0, then Xn converges to X almost surely. Then apply the Borel-Cantelli lemma. You may also use the fact that for any event Bn, {lim supn7oo Bn}° = lim infn70(BM), where we define lim inf Bn := {w EN:W&Bn for finitely many n}. = no