12.4 Show lim sup(sn +tn) < lim sup s, +lim sup tn for bounded sequences (Sn) and (tn). Hint: First show sup{sn + tn : n > N} < sup{sn :n > N}+sup{tn :n > N}.
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- Let {Xn} and {Yn} be sequences of random variables such that Xn diverges to ∞ in probability and Yn is bounded in probability. Show that Xn +Yn diverges to ∞ in probability.(c) Carefully construct a nested family of subsequences (fm,k), and show howthis can be used to produce a single subsequence of (fn) that convergesat every point of A.Define the sequence {Sn} by Sn= [ 1 / (2)(1) ] + [ 1 / (3)(2) ] + ............. + [ 1 / (n+1)(n) ] Prove that Limn→∞ Sn = 1.
- a) Suppose (an) is Cauchy and that for every k ∈ N, the interval (−1/k, 1/k) contains at least one term of (an). Can we say that (an) converges to 0? Either show that it does or give a counter-example.compute the partial sum when k=1 goes until infinity of ak. ak = 1/((k^2)+3k+2)4). Use the Ratio Test to determine whether ∞ n = 1an converges, where an is given. State if the ratio test is inconclusive.∞ n = 134n(n!)4(4n)!Identify an. Evaluate the following limit. lim n → ∞ an+1an Since lim n → ∞ an+1an 1, . please show step by step .