Exercise 3. In this exercise (X, d) is a metric space. .} 2) Assume there exists r> 0 and a sequence (an)neN in the metric space X with d(a,,a,) ≥r, for all i, j EN with ij. Show that X is not compact.
Exercise 3. In this exercise (X, d) is a metric space. .} 2) Assume there exists r> 0 and a sequence (an)neN in the metric space X with d(a,,a,) ≥r, for all i, j EN with ij. Show that X is not compact.
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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![The image contains text from an exercise on metric spaces. Below is the transcription of the visible part of the text for educational purposes:
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**Exercise 3.**
*[In this exercise (X, d) is a metric space.]*
2) Assume there exists \( r > 0 \) and a sequence \( (a_n)_{n \in \mathbb{N}} \) in the metric space \( X \) with \( d(a_i, a_j) \geq r \), for all \( i, j \in \mathbb{N} \) with \( i \neq j \). Show that \( X \) is not compact.
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Note: The rest of the text is obscured and not visible. There are no graphs or diagrams in the visible portion of the image.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9a4dcd82-baf4-45bc-b40b-693a3e683492%2Fbab65394-db75-479f-b40b-1ae7981efab9%2Fciww19_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The image contains text from an exercise on metric spaces. Below is the transcription of the visible part of the text for educational purposes:
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**Exercise 3.**
*[In this exercise (X, d) is a metric space.]*
2) Assume there exists \( r > 0 \) and a sequence \( (a_n)_{n \in \mathbb{N}} \) in the metric space \( X \) with \( d(a_i, a_j) \geq r \), for all \( i, j \in \mathbb{N} \) with \( i \neq j \). Show that \( X \) is not compact.
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Note: The rest of the text is obscured and not visible. There are no graphs or diagrams in the visible portion of the image.
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