S² P, o² and „", → 1, their product converges in *6. In lecture it is claimed that since "S2 probability to o²; in other words, that the product of a convergent sequence of real numbers and a sequence of random variables with a limit in probability converges in probability to the product of the limits. Suppose that {Yn} is a sequence of random variables that converges in probability to Y: lim P (|Yn – Y|> e) = 0 Ve>0 Let {an} be a sequence of real numbers that converges to a. That is, for any e > 0, there is some n* such that n > n* Jan – al < e Show that anYn 24 aY. (Hint: show that an → a and use the property (without proof) that convergence in probability is preserved under multiplication.)

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