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Unformatted text preview: 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 < , x = 0 , x > . 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 , its rather obvious that a > b = > 0 : a b + . (Indeed, if a > b then = a- b obeys the conclusion.) The contrapositive of this statement is logically equivalent, but occasionally useful: > , a < b + = 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, is a least upper bound for S , means two things: (i) s S, s , i.e., is an upper bound for S , i.e., ( * ) s S, s . File sup2009, version of 5 June 2009, page 1. Typeset at 08:59 June 5, 2009. 2 P HILIP D. L OEWEN (ii) Every real number less than is not an upper bound for S . Express this second condition as ( ** ) > , s S : - < s. Notation. A given set S can have at most one least upper bound (LUB). [Pf: Suppose is a least upper bound for S . Any real > breaks ( ** ) use = - and recall ( * ). Any real < breaks ( * )use = - 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 such that = sup S ....
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This note was uploaded on 01/27/2010 for the course MATH 220 taught by Professor Loewen during the Spring '10 term at The University of British Columbia.
- Spring '10