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Suppose b and c are real numbers such that b-c> 1/2. Let {b}and {c}converge to b and N, both b c respectively. Show that there exists a positive integer N such that for all n and c₂ belong to the interval open interval (c—1,b+¹). (Hint: Plot b and c on the number line)

Suppose b and c are real numbers such that b-c> 1/2. Let {b}and {c}converge to b and N, both b c respectively. Show that there exists a positive integer N such that for all n and c₂ belong to the interval open interval (c—1,b+¹). (Hint: Plot b and c on the number line)

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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Suppose \(b\) and \(c\) are real numbers such that \(b < c > \frac{1}{2}\). Let \(\{b_n\}\) and \(\{c_n\}\) converge to \(b\) and \(c\) respectively. Show that there exists a positive integer \(N\) such that for all \(n \geq N\), both \(b_n\) and \(c_n\) belong to the interval open interval \((c - \frac{1}{4}, b + \frac{1}{4})\). (Hint: Plot \(b\) and \(c\) on the number line.) There are no graphs or diagrams provided in this image. The exercise suggests plotting \(b\) and \(c\) on a number line to help visualize the solution.
Transcribed Image Text:Suppose \(b\) and \(c\) are real numbers such that \(b < c > \frac{1}{2}\). Let \(\{b_n\}\) and \(\{c_n\}\) converge to \(b\) and \(c\) respectively. Show that there exists a positive integer \(N\) such that for all \(n \geq N\), both \(b_n\) and \(c_n\) belong to the interval open interval \((c - \frac{1}{4}, b + \frac{1}{4})\). (Hint: Plot \(b\) and \(c\) on the number line.) There are no graphs or diagrams provided in this image. The exercise suggests plotting \(b\) and \(c\) on a number line to help visualize the solution.
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