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12. Suppose (an) is a Cauchy sequence, and that (bn) is a sequence satisfying lim,co lan - bn| = 0. Show that (bn) is a Cauchy sequence.

12. Suppose (an) is a Cauchy sequence, and that (bn) is a sequence satisfying lim,co lan - bn| = 0. Show that (bn) is a Cauchy sequence.

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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**Problem 12: Cauchy Sequences** Suppose \((a_n)\) is a Cauchy sequence, and that \((b_n)\) is a sequence satisfying \(\lim_{n \to \infty} |a_n - b_n| = 0\). Show that \((b_n)\) is a Cauchy sequence. **Explanation:** This problem involves the concept of Cauchy sequences in real analysis. A Cauchy sequence is a sequence where, for any given positive number \(\epsilon\), there exists an integer \(N\) such that for all \(m, n > N\), the distance between the terms \(a_m\) and \(a_n\) is less than \(\epsilon\). The problem asks you to demonstrate that a sequence \((b_n)\), which converges to the sequence \((a_n)\), inherits the Cauchy property when the difference \(|a_n - b_n|\) approaches zero as \(n\) approaches infinity.
Transcribed Image Text:**Problem 12: Cauchy Sequences** Suppose \((a_n)\) is a Cauchy sequence, and that \((b_n)\) is a sequence satisfying \(\lim_{n \to \infty} |a_n - b_n| = 0\). Show that \((b_n)\) is a Cauchy sequence. **Explanation:** This problem involves the concept of Cauchy sequences in real analysis. A Cauchy sequence is a sequence where, for any given positive number \(\epsilon\), there exists an integer \(N\) such that for all \(m, n > N\), the distance between the terms \(a_m\) and \(a_n\) is less than \(\epsilon\). The problem asks you to demonstrate that a sequence \((b_n)\), which converges to the sequence \((a_n)\), inherits the Cauchy property when the difference \(|a_n - b_n|\) approaches zero as \(n\) approaches infinity.
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