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Bekki George Lecture notes 9

Bekki George Lecture notes 9 - Math 1432 Notes Week 9...

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Math 1432 Notes – Week 9 10.1 % THE LEAST UPPER BOUND AXIOM M is an upper bound for S if x M for all x ε S. The least upper bound of S is an upper bound that is less than or equal to any other upper bound for S. LEAST UPPER BOUND AXIOM Every nonempty set of real numbers that has an upper bound has a least upper bound. Examples: 1. {1,2,3,4} S = 2. [?4, 2] 3. ( ) ,8 −∞ 4. ( ) 5, 5. 2 { : 16} S x x = 6. 1 1 1 1 1 1, , , , , , , 2 3 4 5 1000 S = L L THEOREM 10.1.2 If M is the least upper bound of the set S and ε is a positive number, then there is at least one number s in S such that M −ε < s M . Example: 1 2 3 4 , , , , 0.01 2 3 4 5 S ε = = L
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THEOREM 10.1.3 Every nonempty set of real numbers that has a lower bound has a greatest lower bound. Examples: 1. {1,2,3,4} S = 2. [?4, 2] 3. ( ) ,8 −∞ 4. ( ) 5, 5. 2 { : 16} S x x = 6. 1 1 1 1 1 1, , , , , , , 2 3 4 5 1000 S = L L THEOREM 10.1.4 If m is the greatest lower bound of the set S and ε is a positive number, then there is at least one number s in S such that m s < m + ε.
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