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Definition: Let S be a set of real numbers bounded above. A real number b is called an infimum (or greatest lower bound) for S if it has the following two properties: a) b is a lower bound for S. b) No number greater than b is a lower bound for S. If there is a greatest lower bound for S, there is only one and we can speak of the least upper bound. It is denoted as b=inf S and is equal to min S if S has a minimum element.
Source: Mathematical Analysis, second edition by Tom M. Apostol