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1.Suppose that the sequence (an) is monotone. Prove that {an} converges if and only if {a} converges. Show that the result does not hold without the monotonicity assumption.

1.Suppose that the sequence (an) is monotone. Prove that {an} converges if and only if {a} converges. Show that the result does not hold without the monotonicity assumption.

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.Suppose that the sequence {an} is monotone. Prove that {an} converges if and only if {a} converges. Show that the result does not hold without the monotonicity assumption. 2. Suppose that the sequence {a} converges to a and that a < 1. Prove that the sequence {(an)"} converges to zero. (Hint: You need to use the fact that a < 1. You will need to point out in your proof how are using this.)
Transcribed Image Text:1.Suppose that the sequence {an} is monotone. Prove that {an} converges if and only if {a} converges. Show that the result does not hold without the monotonicity assumption. 2. Suppose that the sequence {a} converges to a and that a < 1. Prove that the sequence {(an)"} converges to zero. (Hint: You need to use the fact that a < 1. You will need to point out in your proof how are using this.)
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