Least upper bound
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Given a subset in some larger partially ordered set , a least upper bound or supremum, for is an element such that for every and there is no with this same property.
If the least upper bound of is an element of , it is also the maximum of . If , then has no maximum.
The Least Upper Bound Axiom: This is one of the fundamental axioms of real analysis. According to it, any nonempty set of real numbers that is bounded above has a supremum. This is something intuitively clear but impossible to prove using only the field properties, order properties and completeness property of the set of real numbers.