Question

3. a) Suppose (xn) is a sequence that converges to 0 and (yn) is a bounded sequence. Prove that (XnVn) converges. b) Give an example of a sequence (xn) that converges to x = 0 and a sequence on that is bounded between -1 and 1 such that that (xwyn) does not converge. c) Let xn be any series that converges absolutely and let yn be any series that converges. Prove that xnyn converges absolutely. d) Give an example of a series x, that converges and a series Eyn that converges but Exyn does not converge. e) Let Exy and Eyn be arbitrary series such that Xxx? and Eyn converge. Prove that xn Yn converges absolutely.
Please solve the problem step by step. Thank you very much!

Answer 1

1a). Since (2n) is a bounded sequence So there exist M70 Such that lani <m for all nen. Let Eyo then yo. since So bere exist the sequence (1) Converses to NEN Such that tan1 < am & nys. NOW, 12 An I = 1Xn\\n1 <M. % € - lananl <ɛ & nyn. Hence the Sequence ann) Converges. (b). Consider the Sequen () 1 2 41 Thn An Converges to 27. lel 2n = (-)" then 2 is bearded Now 2mm = 1.17,"fy does not Converse (cy. Since Exis Convergent So exist Kyo such that 1201 As Exn is Convergen so for exist ment Suohoat int, 1946+21+ 1340p < ynym. en b 2 = 1 YDEN. ben A is beaded betwee-,l. In is converged so bounded so toere Such 400t 12.01 < DEN. 3 Convergen so for all Eyo derer • 194ntantilt.. + lanto Antol -lantill Antiltot lantpllAntal <k llantilt. A lantal) TEXE =k. Hence ann CENTCYFS abeculely . Lel. 2n = (Anen Then both the series Inn and Fan are Converges but, => E n 20 = It is divergent.