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 $\phi :{\mathbb{R}}_C\to {\mathbb{R}}_D$, and it fixes the common rationals $\mathbb{Q}$.

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

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