Let X1, X2, .....be a sequence of random variables that converges in probability to a constant a.

Question:

Let X1, X2, .....be a sequence of random variables that converges in probability to a constant a. Assume that P(Xi > 0) = 1 for all i.
a. Verify that the sequences defined by Yi = √Xi and Yii = a/Xi converge in probability.
b. Use the results in part (a) to prove the fact used in Example 5.5.18, that σ/Sn converges in probability to