A) Let {xn } be a bounded sequence. Prove directly using the denition of lim inf and lim sup that there exists two possibly dierent subsequences {xnk } and {xnk } of {xn } where lim inf xk = lim xnk and lim sup xk = lim xnk n k n k 1 1 B) Assume |an | n for n = 1, 2, . . . is a bounded sequence and lim sup |an | n > 0. If R is dened by 1 1 = lim sup |an | n R n then prove that an xn converges for |x| < R. i) the series n=1 an xn diverges for |x| > R. ii) the series n=1 hint: For i) there exists some small > 0 where |x| < R . Now unravel the denition n an xn (for large n) to a of R and what it implies for |an | for large n to compare the series n=1 geometric series! (i.e. use the standard calc 1 comparison test on p. 98 in the book). Note that I want YOU to be more precise than say \"for large n.\" 1 For ii) use problem A) and nd a subsequence ank with R = lim ank . Also there exists small > 0 where |x| > R + . 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 sequence and 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