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Give an example of a metric space X, d and a sequence {Pn} C X such that {pn} is a Cauchy sequence in X,d but {Pn} doesn't converge in X, d (ie X, d is not a complete metric space). Use e-N arguments to show that your sequence is Cauchy and that the limit is not in X, d.

Give an example of a metric space X, d and a sequence {Pn} C X such that {pn} is a Cauchy sequence in X,d but {Pn} doesn't converge in X, d (ie X, d is not a complete metric space). Use e-N arguments to show that your sequence is Cauchy and that the limit is not in X, d.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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**Example of a Metric Space and Cauchy Sequence** **Problem:** Provide an example of a metric space \(X, d\) and a sequence \(\{p_n\} \subseteq X\) such that \(\{p_n\}\) is a Cauchy sequence in \(X, d\) but \(\{p_n\}\) does not converge in \(X, d\) (i.e., \(X, d\) is not a complete metric space). Use \(\varepsilon\)-\(N\) arguments to demonstrate that your sequence is Cauchy and that the limit is not in \(X, d\). **Explanation:** To solve this problem, consider constructing a sequence in a specific metric space that illustrates the properties of Cauchy sequences without reaching convergence due to the incompleteness of the space. You will need to detail your reasoning using \(\varepsilon\)-\(N\) arguments, which involves choosing an appropriate metric space and defining a sequence that meets the criteria.
Transcribed Image Text:**Example of a Metric Space and Cauchy Sequence** **Problem:** Provide an example of a metric space \(X, d\) and a sequence \(\{p_n\} \subseteq X\) such that \(\{p_n\}\) is a Cauchy sequence in \(X, d\) but \(\{p_n\}\) does not converge in \(X, d\) (i.e., \(X, d\) is not a complete metric space). Use \(\varepsilon\)-\(N\) arguments to demonstrate that your sequence is Cauchy and that the limit is not in \(X, d\). **Explanation:** To solve this problem, consider constructing a sequence in a specific metric space that illustrates the properties of Cauchy sequences without reaching convergence due to the incompleteness of the space. You will need to detail your reasoning using \(\varepsilon\)-\(N\) arguments, which involves choosing an appropriate metric space and defining a sequence that meets the criteria.
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