proofs - Completeness Axiom, there exists a supremum, b,...

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Suppose that S is a nonempty set of real numbers that is bounded and that inf(S) = sup(S). Prove that the set S consists of exactly one number. Proof: S is a nonempty set of real numbers that is bounded above and below. Therefor by the
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Unformatted text preview: Completeness Axiom, there exists a supremum, b, and infimum, a. By definition, a <= x for all x in S and b >= x for all x in S. inf(S) <= x <= sup(S) which implies inf(S) = x = sup(S). Therefor there is only one number in the set....
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This note was uploaded on 06/05/2008 for the course MATH 534A taught by Professor Kirschvink during the Spring '08 term at San Diego State.

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