Theorem 1. Suppose F is a complete ordered field. Suppose SF is a nonempty subset which is bounded from below. Then S has an infimum (greatest lower bound). Proof. Consider the set S' = {-x | x € S} of additive inverses. Observe that S' is nonempty and bounded from above. Since F is complete, the set S' has a supremum M. Let m = -M. Then observe that m is the infimum of S. Exercise 1. Complete the above proof by giving detailed justifications for the two observations in the proof.

Theorem 1. Suppose F is a complete ordered field. Suppose SF is a nonempty subset which is bounded from below. Then S has an infimum (greatest lower bound). Proof. Consider the set S' = {-x | x € S} of additive inverses. Observe that S' is nonempty and bounded from above. Since F is complete, the set S' has a supremum M. Let m = -M. Then observe that m is the infimum of S. Exercise 1. Complete the above proof by giving detailed justifications for the two observations in the proof.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter5: Rings, Integral Domains, And Fields

Section5.2: Integral Domains And Fields

Problem 25E: Suppose S is a subset of an field F that contains at least two elements and satisfies both of the...

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