# lec8 - Math 117 – 8 The completeness axiom Definition 8.1...

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Unformatted text preview: Math 117 – April 21, 2011 8 The completeness axiom Definition 8.1. Let S ⊆ R . If there exists a real number M , such that M ≥ s for all s ∈ S , then M is called an upper bound for S , and we say that S is bounded above. If there exists a real number m , such that m ≤ s for all s ∈ S , then m is called a lower bound for S , and we say that S is bounded below. A set is called bounded, if it is bounded above and below. If M ∈ S is an upper bound for S , then M is called the maximum of S , M = max S , and if m ∈ S is a lower bound for S , then m is called the minimum of S , m = min S . Definition 8.2. Let S ⊆ R be nonempty. If S is bounded above, then the least upper bound is called the supremum of S , denoted by sup S . If S is bounded below, then the greatest lower bound is called its infimum, denoted by inf S . Thus, M = sup S is equivalent to the conditions (similar conditions hold for the infimum with reversed inequalities) ( a ) m ≥ s for all s ∈ S ( b ) If m ≥...
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