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.
Completeness: This is one of the fundamental axioms of real analysis.
A set is said to be complete if any nonempty subset of that is bounded above has a supremum.
The fact that is complete is something intuitively clear but impossible to prove using only the field and order properties of