This exercise, and the two following, will look at some of the mathematical details of convergence. (a)

Question:

This exercise, and the two following, will look at some of the mathematical details of convergence.
(a) Prove Theorem 5.5.4. (Hint: Since h is continuous, given ∈ > 0 we can find a S such that |h(xn) - h(x)\< ∈ whenever |xn - x| < δ. Translate this into probability statements.)
(b) In Example 5.5.8, find a subsequence of the XiS that converges almost surely, that is, that converges pointwise.