Unformatted text preview: a âˆˆ A . This shows thatÎ· is an upper bound forA . Suppose Î² â‰¤ Î· . IfÎ· is to be the least upper bound forA , we must show that there exists an elementa âˆˆ A witha â‰¥ Î² . If Î² â‰¤ Î· , thenÎ² â‰¥ Î· . Since Î· is the greatest lower bound for A , it follows thatÎ² is not a lower bound for A . Thus, there exists an a âˆˆ A with a â‰¤ Î² . I.e.,a â‰¥ Î² . Sincea âˆˆ A when a âˆˆ A , this completes the proof. 1...
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 Spring '10
 MetCalfe
 Calculus, Logic, Empty set, Supremum, Order theory, upper bound

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