sn6_1 - Math 1432 Notes Week 6 Note: The exam covers...

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Math 1432 Notes – Week 6 Note: The exam covers chapters 7-9! Tonight’s material is NOT on the midterm. We will review at the end of class tonight. 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 = m m
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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 = = m 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
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sn6_1 - Math 1432 Notes Week 6 Note: The exam covers...

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