Let (an) and (bn) be bounded real sequences. Using the definitions (a) lim sup, → an = lim, [sup>n ar] and lim inf,∞ An = lim, infr2n ak] %3D n→∞ (b) Show that lim sup(a, + bn) < lim sup an + lim sup b, and lim inf(a, + b,) > lim inf an + lim infb, . You may use well known properties of sup and inf. (c) For each inequality, find an example to show that equality doesn't always hold.
Let (an) and (bn) be bounded real sequences. Using the definitions (a) lim sup, → an = lim, [sup>n ar] and lim inf,∞ An = lim, infr2n ak] %3D n→∞ (b) Show that lim sup(a, + bn) < lim sup an + lim sup b, and lim inf(a, + b,) > lim inf an + lim infb, . You may use well known properties of sup and inf. (c) For each inequality, find an example to show that equality doesn't always hold.
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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![Let (an) and (bn) be bounded real sequences. Using the definitions
(а)
lim
supn+0
An
= lim (sup>n ak] and lim inf→ an = lim,+0 infr>n ak]
(b) Show that lim sup(an + bn) < lim sup an + lim sup b, and lim inf(a, + br) > lim inf an + lim inf b, . You
may use well known properties of sup and inf.
(c) For each inequality, find an example to show that equality doesn't always hold.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4703214b-7b6d-48dc-8fec-73bc2badaa48%2F30440f83-7701-4984-8983-f4c9f6f69764%2F0hji3dn_processed.png&w=3840&q=75)
Transcribed Image Text:Let (an) and (bn) be bounded real sequences. Using the definitions
(а)
lim
supn+0
An
= lim (sup>n ak] and lim inf→ an = lim,+0 infr>n ak]
(b) Show that lim sup(an + bn) < lim sup an + lim sup b, and lim inf(a, + br) > lim inf an + lim inf b, . You
may use well known properties of sup and inf.
(c) For each inequality, find an example to show that equality doesn't always hold.
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