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• Exercise 2.1.18. Let {n} be a sequence, and let a be a real number. Suppose that for every positive &, there is an M such that n − x ≤ e when n ≥ M. Show that limn→∞ In = x.

• Exercise 2.1.18. Let {n} be a sequence, and let a be a real number. Suppose that for every positive &, there is an M such that n − x ≤ e when n ≥ M. Show that limn→∞ In = x.

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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Please solve exercise 2.1.18, with explanations... all information for the question is provided

• Exercise 2.1.18. Let {n} be a sequence, and let a be a real number. Suppose that for every positive , there is an M such that n - x ≤e when n ≥ M. Show that limn→∞ n = x. Exercise 2.2.5. Suppose n - cos(n) In := n Use the squeeze lemma to show that the sequence {n} converges, and find the limit. Remarks The statement in Exercise 2.1.18 is almost the same as Definition 2.1.2, except that "< " has been replaced by the apparently weaker property "< €." Your task is to explain why this change actually does not matter. In Exercise 2.2.5, the instruction to "find the limit" is not really a separate task, for the conclusion of the squeeze lemma identifies the value of the limit.
Transcribed Image Text:• Exercise 2.1.18. Let {n} be a sequence, and let a be a real number. Suppose that for every positive , there is an M such that n - x ≤e when n ≥ M. Show that limn→∞ n = x. Exercise 2.2.5. Suppose n - cos(n) In := n Use the squeeze lemma to show that the sequence {n} converges, and find the limit. Remarks The statement in Exercise 2.1.18 is almost the same as Definition 2.1.2, except that "< " has been replaced by the apparently weaker property "< €." Your task is to explain why this change actually does not matter. In Exercise 2.2.5, the instruction to "find the limit" is not really a separate task, for the conclusion of the squeeze lemma identifies the value of the limit.
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