Equivalence of the Cauchy and Dedekind constructions of $\mathbb{R}$

$\mathbb{R}_C$ is a complete ordered field: a totally ordered field ([L1]) with the least-upper-bound property ([L2]).

$\mathbb{R}_D$ is a complete ordered field: a totally ordered field ([L3]) with the least-upper-bound property ([L4]).

By [L5] applied to $F = \mathbb{R}_C$ and $G = \mathbb{R}_D$ there is a unique ordered-field isomorphism $\varphi : \mathbb{R}_C \to \mathbb{R}_D$, and it fixes the common rationals $\mathbb{Q}$.

As a field isomorphism $\varphi$ preserves $+$, $\cdot$, $0$, $1$ and inverses; as an ordered-field isomorphism it satisfies $x < y \iff \varphi x < \varphi y$, hence preserves $\le$ and $|\cdot|$; and it preserves suprema, in the sense that for any nonempty $S \subseteq \mathbb{R}_C$ bounded above with $s = \sup S$, its image $\varphi[S] = \{\varphi(t) : t \in S\}$ has $\varphi(s) = \sup \varphi[S]$, since $\varphi(s)$ is an upper bound of $\varphi[S]$ and, $\varphi^{-1}$ being order-preserving, every upper bound of $\varphi[S]$ is $\ge \varphi(s)$.

Therefore $\mathbb{R}_C$ and $\mathbb{R}_D$ are the same complete ordered field presented two ways, joined by the unique isomorphism $\varphi$ that restricts to the identity on $\mathbb{Q}$ and preserves all arithmetic and order: the Cauchy and Dedekind constructions give the same $\mathbb{R}$.