SupInfSetsSummary10f

SupInfSetsSummary10f - Supremum and Infimum of a Set...

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Unformatted text preview: Supremum and Infimum of a Set Summary Set up • S ⊂ R • R is the set of real numbers • b R := R ∪ {∞} ∪ {-∞} is the set of extended real numbers Supremum of S = sup S also called Least Upper Bound of S = lub S Def.’s • Let S be a nonempty set that is bounded above. Then β ∈ R is a sup S provided (1) β is an upper bound of S (i.e., ∀ x ∈ S , x ≤ β ) (2) if b < β , then b is not an upper bound of S . • Let S be a nonempty set that is not bounded above. Then sup S := ∞ . • sup ∅ :=- ∞ . Thm. Let S be a nonempty set that is bounded above. Then the sup S is the unique real number β ∈ R such that (1) β is an upper bound of S (i.e., ∀ x ∈ S , x ≤ β ) (2) if b < β , then b is not an upper bound of S (2 ) if b is an upper bound of S , then β ≤ b (2 00 ) if b < β , then ∃ x b ∈ S such that b < x b (2 000 ) if ε > 0, then ∃ x ε ∈ S such that β- ε < x ε (2 0000 ) if ε > 0, then ∃ x ε ∈ S such that β- 17 ε < x ε ....
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