Question: Suppose that the sequence (X) is fundamental in probability in the sense that for e positive there exists an N such that P[IX-X,>e] N.

Suppose that the sequence (X) is fundamental in probability in the sense that for e positive there exists an N such that P[IX-X,>e] N. (a) Prove there is a subsequence (X) and a random variable X such that lim, XX with probability 1. Hint: Choose increasing n such that P[IX-X>2-k] 2-k]. nk m (b) Show that Xp X. n

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