The Real Numbers Form an Ordered Field in Which Every Nonempty Set Bounded Above Has a Supremum

In the setting of The Integers and the Rational Numbers, with the Natural Numbers and the Integers Identified with Subsets of the Rationals, let $\mathbb{R}$, $\le$, $+$, $\cdot$, $0_{\mathbb{R}}$ and $1_{\mathbb{R}}$ be as in The Real Numbers §reals, The Real Numbers §operations and The Real Numbers §constants, and $u^{*}$ for $u\in\mathbb{Q}$ as in Construction of the Dedekind Cuts: They Form a Set Totally Ordered by Inclusion, Closed under Sums, Negatives and Products of Nonnegative Cuts.

¶ $\mathbb{R}$, with $+$, $\cdot$, $0_{\mathbb{R}}$, $1_{\mathbb{R}}$ and $\le$, is an ordered field.

¶ Every subset of $\mathbb{R}$ other than the empty set that is bounded above has a supremum.

¶ The canonical embedding $\kappa_{\mathbb{R}}$ of $\mathbb{Q}$ into the ordered field $\mathbb{R}$ of clause ordered-field satisfies $\kappa_{\mathbb{R}}(u)=u^{*}$ for every $u\in\mathbb{Q}$.