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Every nonempty set of real numbers that is bounded above has a least upper bound.
Completeness axiom (least upper bound property): If is nonempty and bounded above, then has a supremum in .
This axiom distinguishes from other ordered fields and underlies many foundational results, including the monotone sequence convergence theorem and the equivalences collected in completeness equivalences.
A bounded above set is a subset for which there exists such that for every ; such an is called an upper bound of .
A supremum of a nonempty set that is bounded above is a real number such that:
The supremum is the “least upper bound” and may exist even when has no maximum. The completeness axiom asserts that every nonempty bounded-above set of real numbers has a supremum.
Monotone sequence convergence theorem: Let be a monotone sequence of real numbers.
Completeness equivalences: For an ordered field satisfying the field axioms and order axioms, the following statements are equivalent (each implies the others):