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2. Let {an} be a monotone sequence. Show that {a,} is a Cauchy sequence if and only if it is bounded.

2. Let {an} be a monotone sequence. Show that {a,} is a Cauchy sequence if and only if it is bounded.

Advanced Engineering Mathematics
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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1. Let a 1 and define an+1 = /4a, - 1 for n 2 1. Show that the sequence {an} converges and find the limit. 2. Let {an} be a monotone sequence. Show that {a,} is a Cauchy sequence if and only if it is bounded. 3. Consider the sequence an = sin n = 1, 2, 3, 4, .. Find lim sup an, lim inf an. n00 4. Find the following limits: x2 + x – 6 lim |a - 2| x2 - 4 lim x+2- x - x-2 4 Note that the second limit above is a one-sided limit! 5. Use the ɛ - 8 definition to show that the function x - 1 f(x) = x+2 is continuous at x = 3. x* has at least one real root. You 6. Show that the equation 5ª = may assume that the functions 5ª and x are both continuous on (-00, +0).
Transcribed Image Text:1. Let a 1 and define an+1 = /4a, - 1 for n 2 1. Show that the sequence {an} converges and find the limit. 2. Let {an} be a monotone sequence. Show that {a,} is a Cauchy sequence if and only if it is bounded. 3. Consider the sequence an = sin n = 1, 2, 3, 4, .. Find lim sup an, lim inf an. n00 4. Find the following limits: x2 + x – 6 lim |a - 2| x2 - 4 lim x+2- x - x-2 4 Note that the second limit above is a one-sided limit! 5. Use the ɛ - 8 definition to show that the function x - 1 f(x) = x+2 is continuous at x = 3. x* has at least one real root. You 6. Show that the equation 5ª = may assume that the functions 5ª and x are both continuous on (-00, +0).
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