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Which of the following defines a Cauchy sequence in a metric space (M,d)? O Ve>0 (3kZ (Vm, n>k (d (m, n) < €))) (V> (3kZ+ (vn >k (d (an, l) < €)))) A convergent sequence is a Cauchy sequence. A sequence that does not converge is a Cauchy sequence.

Which of the following defines a Cauchy sequence in a metric space (M,d)? O Ve>0 (3kZ (Vm, n>k (d (m, n) < €))) (V> (3kZ+ (vn >k (d (an, l) < €)))) A convergent sequence is a Cauchy sequence. A sequence that does not converge 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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**Question:** Which of the following defines a Cauchy sequence in a metric space (M, d)? 1. \( \forall \epsilon > 0 \left( \exists k \in \mathbb{Z}^+ \left( \forall m, n > k \left( d (x_m, x_n) < \epsilon \right) \right) \right) \) 2. \( \exists l \in M (\forall \epsilon > 0 \left( \exists k \in \mathbb{Z}^+ \left( \forall n > k \left( d (x_n, l) < \epsilon \right) \right) \right)) \) 3. A convergent sequence is a Cauchy sequence. 4. A sequence that does not converge is a Cauchy sequence. **Explanation of Options:** - **Option 1** describes the formal definition of a Cauchy sequence. It states that for every positive epsilon, there exists a positive integer \( k \) such that for all indices \( m, n \) greater than \( k \), the distance between \( x_m \) and \( x_n \) is less than epsilon. - **Option 2** describes a convergent sequence condition, where there exists an element \( l \) in the metric space \( M \), such that for every positive epsilon, there is a positive integer \( k \) where the distance from \( x_n \) to \( l \) is less than epsilon for all indices \( n \) greater than \( k \). - **Option 3** states a mathematical fact: every convergent sequence in a metric space is a Cauchy sequence. - **Option 4** is incorrect as it states a false proposition. In a complete metric space, every Cauchy sequence converges, so a sequence that does not converge cannot be a Cauchy sequence.
Transcribed Image Text:**Question:** Which of the following defines a Cauchy sequence in a metric space (M, d)? 1. \( \forall \epsilon > 0 \left( \exists k \in \mathbb{Z}^+ \left( \forall m, n > k \left( d (x_m, x_n) < \epsilon \right) \right) \right) \) 2. \( \exists l \in M (\forall \epsilon > 0 \left( \exists k \in \mathbb{Z}^+ \left( \forall n > k \left( d (x_n, l) < \epsilon \right) \right) \right)) \) 3. A convergent sequence is a Cauchy sequence. 4. A sequence that does not converge is a Cauchy sequence. **Explanation of Options:** - **Option 1** describes the formal definition of a Cauchy sequence. It states that for every positive epsilon, there exists a positive integer \( k \) such that for all indices \( m, n \) greater than \( k \), the distance between \( x_m \) and \( x_n \) is less than epsilon. - **Option 2** describes a convergent sequence condition, where there exists an element \( l \) in the metric space \( M \), such that for every positive epsilon, there is a positive integer \( k \) where the distance from \( x_n \) to \( l \) is less than epsilon for all indices \( n \) greater than \( k \). - **Option 3** states a mathematical fact: every convergent sequence in a metric space is a Cauchy sequence. - **Option 4** is incorrect as it states a false proposition. In a complete metric space, every Cauchy sequence converges, so a sequence that does not converge cannot be a Cauchy sequence.
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