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Definition: Let S be a set of real numbers bounded above. A real number b is called a supremum (or least upper bound) for S if it has the following two properties: a) b is an upper bound for S. b) No number less than b is an upper bound. If there is a least upper bound for S, there is only one and it is denoted as b=sup S. If S has a maximum element, then max S=sup S.
Source: Mathematical Analysis, second edition by Tom M. Apostol