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space is the union of a finite number of closed balls radius e. Prove that a metric space is totally bounded if and only if every sequence has a Cauchy subsequence.

space is the union of a finite number of closed balls radius e. Prove that a metric space is totally bounded if and only if every sequence has a Cauchy subsequence.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter7: Distance And Approximation
Section7.2: Norms And Distance Functions
Problem 14EQ
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1 Call a metric space totally bounded if, for every e > 0, the metric space is the union of a finite number of closed balls radius e. Prove that a metric space is totally bounded if and only if every sequence has a Cauchy subsequence.
Transcribed Image Text:1 Call a metric space totally bounded if, for every e > 0, the metric space is the union of a finite number of closed balls radius e. Prove that a metric space is totally bounded if and only if every sequence has a Cauchy subsequence.
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