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For example, suppose A is R2. Then the set N = {(m, n) | m, n = 0, ±1, ...} constitutes an e-net for R² (see Fig. 5.1), provided & is greater than √2/2. Why is this so? What is the significance?

For example, suppose A is R2. Then the set N = {(m, n) | m, n = 0, ±1, ...} constitutes an e-net for R² (see Fig. 5.1), provided & is greater than √2/2. Why is this so? What is the significance?

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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DEFINITION 5.6. The set of points N of the metric space (x, d) is said to be an ɛ-net with respect to the set A if, for any xe A, there exists a yeN such that -2 -1 1 2 23 FIG. 5.1. An &-net for R² for & >√√/2/2. For example, suppose A is R2. Then the set N = {(m, n) | m, n = 0, ±1, ...} constitutes an e-net for R² (see Fig. 5.1), provided & is greater than √√2/2. Why is this so? What is the significance?
Transcribed Image Text:DEFINITION 5.6. The set of points N of the metric space (x, d) is said to be an ɛ-net with respect to the set A if, for any xe A, there exists a yeN such that -2 -1 1 2 23 FIG. 5.1. An &-net for R² for & >√√/2/2. For example, suppose A is R2. Then the set N = {(m, n) | m, n = 0, ±1, ...} constitutes an e-net for R² (see Fig. 5.1), provided & is greater than √√2/2. Why is this so? What is the significance?
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