Suppose that ACR is bounded below, and define B = {b € R | b is a lower bound for A}. Prove that sup B = inf A. Use (a) to explain why there is no need to assert that greatest upper bounds exist as part of the Axiom of Completeness.

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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Suppose that ACR is bounded below, and define B
b is a lower bound for A}. Prove that sup B
=
inf A.
=
{b € R |
Use (a) to explain why there is no need to assert that greatest upper
bounds exist as part of the Axiom of Completeness.
Transcribed Image Text:Suppose that ACR is bounded below, and define B b is a lower bound for A}. Prove that sup B = inf A. = {b € R | Use (a) to explain why there is no need to assert that greatest upper bounds exist as part of the Axiom of Completeness.
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