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THEOREM 11.1.2 If M is the least upper bound of the set S and is a positive number, then there is at least one number s in S such that M- 0. Since M is an upper bound for S, the conditions

THEOREM 11.1.2 If M is the least upper bound of the set S and is a positive number, then there is at least one number s in S such that M- 0. Since M is an upper bound for S, the conditions

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter1: Vectors
Section1.1: The Geometry And Algebra Of Vectors
Problem 24EQ
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Prove 11.1.2

THEOREM 11.1.2 If M is the least upper bound of the set S and is a positive number, then there is at least one number s in S such that M-<s≤M. PROOF Let > 0. Since M is an upper bound for S, the conditions <M is satisfied by all numbers s in S. All we have to show therefore is that there is some number s in S such that M-Є<s.
Transcribed Image Text:THEOREM 11.1.2 If M is the least upper bound of the set S and is a positive number, then there is at least one number s in S such that M-<s≤M. PROOF Let > 0. Since M is an upper bound for S, the conditions <M is satisfied by all numbers s in S. All we have to show therefore is that there is some number s in S such that M-Є<s.
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