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sup2009

# sup2009 - Supremum and Infimum UBC M220 Lecture Notes by...

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Supremum and Infimum UBC M220 Lecture Notes by Philip D. Loewen The Real Number System. Work hard to construct from the axioms a set R with special elements O and I , and a subset P R , and mappings A : R × R R , M : R × R R , for which defining the basic operations above in terms of x + y = A ( x, y ) , x · y = M ( x, y ) , x > O x P produces a consistent setup in which the familiar rules of arithmetic all work. Trichotomy. For every real number x , exactly one of the following is true: x < 0 , x = 0 , x > 0 . By taking x = b - a , we deduce that whenever a, b R , exactly one of the following is true: a < b, a = b, a > b. Given a, b R , it’s rather obvious that a > b = ⇒ ∃ ε > 0 : a b + ε. (Indeed, if a > b then ε = a - b obeys the conclusion.) The contrapositive of this statement is logically equivalent, but occasionally useful: ε > 0 , a < b + ε = a b. It reveals that one way to prove “ a b ” is to prove a collection of apparently easier inequalities involving a larger right-hand side. Definition. Let S R . To say, “ S is bounded above,” means there exists b R such that ( * ) s S, s b. Any number b satisfying ( * ) is called “an upper bound for S .” Changing “ ” to “ ” in ( * ) produces a definition for the phrases “ S is bounded below” and “ b is a lower bound for S ”. To say that S is bounded means that S is bounded above and S is bounded below. Definition. Let S R . The phrase, “ β is a least upper bound for S ,” means two things: (i) s S, s β , i.e., β is an upper bound for S , i.e., ( * ) s S, s β. File “sup2009”, version of 5 June 2009, page 1. Typeset at 08:59 June 5, 2009.

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2 P HILIP D. L OEWEN (ii) Every real number less than β is not an upper bound for S . Express this second condition as ( ** ) ε > 0 , s S : β - ε < s. Notation. A given set S can have at most one least upper bound (LUB). [Pf: Suppose β 0 is a least upper bound for S . Any real β > β 0 breaks ( ** )— use ε = β - β 0 and recall ( * ). Any real β < β 0 breaks ( * )—use ε = β 0 - β and recall ( ** ).] If S has a least upper bound, it is a unique element of R denoted sup S (Latin supremum ). A symmetric development leads to the concepts of sets bounded below, greatest lower bounds, and the Latin notation inf S (Latin infimum ). The Least Upper Bound Property. The hard work in the axiomatic construction of the real number system is in arranging the following fundamental property: For each nonempty subset S of R , if S has an upper bound, then there exists a unique real number β such that β = sup S .
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sup2009 - Supremum and Infimum UBC M220 Lecture Notes by...

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