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14. Suppose (an) is a bounded sequence such that all of its converging subsequences converge to the same limit, say L. Show that (an) converges to L as

14. Suppose (an) is a bounded sequence such that all of its converging subsequences converge to the same limit, say L. Show that (an) converges to L as

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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**14. Suppose \((a_n)\) is a bounded sequence such that all of its converging subsequences converge to the same limit, say \(L\). Show that \((a_n)\) converges to \(L\) as well.** In this problem, you are given a sequence \((a_n)\) which is bounded. This means there exists some positive number \(M\) such that for all terms in the sequence, \(|a_n| \leq M\). A subsequence is a sequence derived from another sequence by deleting some or no elements without changing the order of the remaining elements. The condition given here is that if any subsequence of \((a_n)\) converges, it converges to the same limit \(L\). The task is to prove that the original sequence \((a_n)\) itself converges to \(L\). This exercise typically involves using properties of bounded sequences and understanding the concept of convergence and subsequential limits.
Transcribed Image Text:**14. Suppose \((a_n)\) is a bounded sequence such that all of its converging subsequences converge to the same limit, say \(L\). Show that \((a_n)\) converges to \(L\) as well.** In this problem, you are given a sequence \((a_n)\) which is bounded. This means there exists some positive number \(M\) such that for all terms in the sequence, \(|a_n| \leq M\). A subsequence is a sequence derived from another sequence by deleting some or no elements without changing the order of the remaining elements. The condition given here is that if any subsequence of \((a_n)\) converges, it converges to the same limit \(L\). The task is to prove that the original sequence \((a_n)\) itself converges to \(L\). This exercise typically involves using properties of bounded sequences and understanding the concept of convergence and subsequential limits.
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