1. Recall that for a bounded sequence {xk}, we definedinf (sup x; ...}].j>klim sup xkk>1(i) Show that if M > lim supëk,then there exists N such that n > N =→Xn < M.(ii) Show that if m < lim sup,then there exists infinitely many n so thatXn 2 m.

We have defined. limk→∞sup xk=infk≥1[supj≥k xj]...........(1)

Answered: 1. Recall that for a bounded sequence…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Q8 A
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1. Recall that for a bounded sequence {xk}, we defined inf [sup a, ...}]. k21°j>k lim sup xk = k→∞ (i) Show that if M > lim suprk, then there exists N such that n > N → Xn < M. (ii) Show that if m < lim sup, then there exists infinitely many n so that Xn 2 m.