A) Let {xn } be a bounded sequence. Prove directly using the definition of lim inf and lim supthat there exists two possibly different subsequences {xnk } and {xnk } of {xn } wherelim inf xk = lim xnk...

A) Let {xn } be a bounded sequence. Prove directly using the definition of lim inf and lim supthat there exists two possibly different subsequences {xnk } and {xnk } of {xn } wherelim inf xk = lim xnk and lim sup xk = lim xnkn→∞k→∞n→∞k→∞11B) Assume |an | n for n = 1, 2, . . . is a bounded sequence and lim sup |an | n > 0. If R is definedby11= lim sup |an | nRn→∞then prove that∞an xn converges for |x| <> R.ii) the seriesn=1hint: For i) there exists some small δ > 0 where |x| <> 0 where |x| > R + δ. What can you then say about lim ank |x|nk ?k→∞C) Prove Theorem 3.7.8 on p. 99 under the assumption that {|xn /yn |} is a bounded sequenceand where r is replaced by lim inf |xn /yn |.Also on p. 110 do: 3, 4, 10, 13.I will collect: A), B), and 10, 13 on p.110.