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.

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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.
0
Exercise 1. Complete the above proof by giving detailed justifications for
the two observations in the proof.
Transcribed Image Text: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. 0 Exercise 1. Complete the above proof by giving detailed justifications for the two observations in the proof.
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