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For each case, determine whether the metric space is complete or not. Support your answers. (i) Z with the usual metric. (ii) (R, d), where d(x, y) = min{|x − y\, 1}. (0-0) Xn Yn (iii) (R", d), where (iv) (N, d), where d(m, n) = | / - // | - 1 n m = max{|æk — yk|: k = 1, ..., n}.

For each case, determine whether the metric space is complete or not. Support your answers. (i) Z with the usual metric. (ii) (R, d), where d(x, y) = min{|x − y\, 1}. (0-0) Xn Yn (iii) (R", d), where (iv) (N, d), where d(m, n) = | / - // | - 1 n m = max{|æk — yk|: k = 1, ..., n}.

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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For each case, determine whether the metric space is complete or not. Support your answers. (i) \(\mathbb{Z}\) with the usual metric. (ii) \((\mathbb{R}, d)\), where \(d(x, y) = \min\{|x - y|, 1\}\). (iii) \((\mathbb{R}^n, d)\), where \(d\left(\begin{pmatrix} x_1 \\ \vdots \\ x_n \end{pmatrix}, \begin{pmatrix} y_1 \\ \vdots \\ y_n \end{pmatrix}\right) = \max\{|x_k - y_k| : k = 1, \ldots, n\}\). (iv) \((\mathbb{N}, d)\), where \(d(m, n) = \left|\frac{1}{n} - \frac{1}{m}\right|\).
Transcribed Image Text:For each case, determine whether the metric space is complete or not. Support your answers. (i) \(\mathbb{Z}\) with the usual metric. (ii) \((\mathbb{R}, d)\), where \(d(x, y) = \min\{|x - y|, 1\}\). (iii) \((\mathbb{R}^n, d)\), where \(d\left(\begin{pmatrix} x_1 \\ \vdots \\ x_n \end{pmatrix}, \begin{pmatrix} y_1 \\ \vdots \\ y_n \end{pmatrix}\right) = \max\{|x_k - y_k| : k = 1, \ldots, n\}\). (iv) \((\mathbb{N}, d)\), where \(d(m, n) = \left|\frac{1}{n} - \frac{1}{m}\right|\).
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