please solve part B on page

Exercise 4. a) Let (an, n 2 1), (bn, n 2 1) be sequences of real numbers with an+1 2 an > 0 for every n 2 1 and limn too an = too. Show that n bk n if lim exists, then lim bk = 0. n-+0c ak k=1 n-+0o un kzl Hint: Use the following lemma (whose proof is not required): n if lim Un = v, then lim UK (ak - ak-1) = v; 1-+00 n-+co an k=1 and use Ck_1 bk = ER_1 ak(Vk - VK-1) with Un := 2k=1 ak b) Let (Xn, n 2 1) be a sequence of independent random variables with E(Xn) = 0 for every n 2 1. Show that n Mn := ) Xk k= 1 k is a martingale with respect to the filtration Fn := o(X1, ..., Xn). c) If supn>1 E(X2) ] E(Y?)