20.21. 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 for m, nzn. Analyze P[IX-Xn>2-k]. (b) Show that X₁ →p X.

A jar contains n red balls labeled from 1 to n where n ≥ 2 and m blue balls. a. Two balls are selected at random, one after another, from the jar without replacement. What is the probability that the pair of ball is red? b. One red ball is selected at random from the jar, let pn be the probability of selecting a red ball labeled with a prime even number. Find pn and then compute lim Pn. n-∞

2. Suppose the sequence {sn} satisfies |8n+1 = 8n| ≤ 72²2 for all n E N. Prove that {n} converges.

2- Suppose that (xn) is a sequence such that xn > 0 for all NE N. 1 = 0. Show that; Xn → 0 as n → 0∞ lim n→o Xn O lin I0 - Show also; Xns-0∞ as n