Complete ordered field (least-upper-bound property)

Let $F$ be an ordered field (Ordered field) and $S\subseteq F$.

• $u\in F$ is an upper bound of $S$ if $s\le u$ for all $s\in S$; $S$ is bounded above if it has an upper bound.
• $u\in F$ is a least upper bound (or supremum, $\sup S$) of $S$ if $u$ is an upper bound of $S$ and $u\le u^{\mathrm{\prime }}$ for every upper bound $u^{\mathrm{\prime }}$ of $S$.

$F$ is a complete ordered field (equivalently, $F$ has the least-upper-bound property, or is Dedekind complete) if every nonempty $S\subseteq F$ that is bounded above has a least upper bound in $F$.

Ordered field

• A least upper bound, if it exists, is unique (two least upper bounds are each $\le$ the other, so equal by antisymmetry of the order).
• Applying the property to $\{-s:s\in S\}$ yields the dual greatest lower bound (infimum) property, so the two are equivalent.
• The Dedekind-cut reals have this property by construction (Dedekind completeness: the least-upper-bound property); the Cauchy-sequence reals acquire it via The Cauchy-sequence reals have the least-upper-bound property. This definition is the target for the uniqueness theorem Uniqueness of the complete ordered field: $\mathbb{R}$ up to a unique isomorphism.