Chapter 2.5, Problem 16ES

i. Suppose that the sn satisfies both limn→∞ s2n = 3 and limn→∞ s2n+1 = 3. (That is, the sequence given by the even terms of sn and that given by the odd terms of sn both converge to 3.) Show that also limn→∞ sn = 3.ii. Give an example of a sequence where the sequences given by the even and by the odd terms both converge, but where the entire sequence does not converge.

Suppose that the sn satisfies both limn→∞ s2n = 2 and limn→∞ s2n+1 = 2. (That is, the sequence given by the even terms of sn and that given by the odd terms of sn both converge to 2.) Show that also limn→∞ sn = 2.

Suppose that we observe that X1, X2, . . . , Xn  are iid∼ U(0, 1). Show that X(1)converges in probability to zero.