Addition and Order of Dedekind Cuts: an Ordered Abelian Group 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 $C$, $\le_{C}$, $u^{*}$ for $u\in\mathbb{Q}$, and $\ominus$ be as in Construction of the Dedekind Cuts: They Form a Set Totally Ordered by Inclusion, Closed under Sums, Negatives and Products of Nonnegative Cuts, and let $+$ and $0_{\mathbb{R}}$ be as in The Real Numbers §operations and The Real Numbers §constants (so that $\mathbb{R}=C$, by The Real Numbers §reals); bounds and suprema of subsets of $C$ are taken with respect to the total order $\le_{C}$ (Construction of the Dedekind Cuts: They Form a Set Totally Ordered by Inclusion, Closed under Sums, Negatives and Products of Nonnegative Cuts §order), as in Bounds, Least and Greatest Elements, Suprema and Infima for a Partial Order. Let $x,y,z\in C$ and $u,v\in\mathbb{Q}$.

¶ $(x+y)+z=x+(y+z)$, $x+y=y+x$, $x+0_{\mathbb{R}}=x$ and $x+\ominus x=0_{\mathbb{R}}$.

¶ If $x\le_{C}y$, then $x+z\le_{C}y+z$.

¶ If $s$ is a subset of $C$ other than the empty set that is bounded above, then its union $\bigcup s$ lies in $C$ and is the supremum of $s$.

¶ $(u+v)^{*}=u^{*}+v^{*}$.

¶ $u\le v$ if and only if $u^{*}\le_{C}v^{*}$.