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

By The Real Numbers §reals, $\mathbb{R}=C$ with $C$ 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 $\le$ is $\le_{C}$; $\ominus$ is as in that lemma. $\mathbb{R}$ is a set by Construction of the Dedekind Cuts: They Form a Set Totally Ordered by Inclusion, Closed under Sums, Negatives and Products of Nonnegative Cuts §set; $+$ and $\cdot$ are binary operations on $\mathbb{R}$ by The Real Numbers §operations; $0_{\mathbb{R}},1_{\mathbb{R}}\in\mathbb{R}$ by The Real Numbers §constants; and $\le$ is a total order on $\mathbb{R}$ by Construction of the Dedekind Cuts: They Form a Set Totally Ordered by Inclusion, Closed under Sums, Negatives and Products of Nonnegative Cuts §order. Let $x,y,z\in\mathbb{R}$.

Clause ordered-field. Commutative ring. The identities $(x+y)+z=x+(y+z)$, $x+y=y+x$ and $x+0_{\mathbb{R}}=x$ hold by Addition and Order of Dedekind Cuts: an Ordered Abelian Group in Which Every Nonempty Set Bounded Above Has a Supremum §group, and $(x\cdot y)\cdot z=x\cdot(y\cdot z)$, $x\cdot y=y\cdot x$, $x\cdot1_{\mathbb{R}}=x$ and $x\cdot(y+z)=x\cdot y+x\cdot z$ hold by Multiplication of Dedekind Cuts: Associativity, Commutativity, Unit, Distributivity, Reciprocals and Nonnegative Products §ring. For the additive inverse take $w=\ominus x$, which lies in $\mathbb{R}$ by Construction of the Dedekind Cuts: They Form a Set Totally Ordered by Inclusion, Closed under Sums, Negatives and Products of Nonnegative Cuts §negative and satisfies $x+w=0_{\mathbb{R}}$ by Addition and Order of Dedekind Cuts: an Ordered Abelian Group in Which Every Nonempty Set Bounded Above Has a Supremum §group. So $\mathbb{R}$, with $+$, $\cdot$, $0_{\mathbb{R}}$ and $1_{\mathbb{R}}$, is a commutative ring.

Field. $0_{\mathbb{R}}\neq1_{\mathbb{R}}$ by Multiplication of Dedekind Cuts: Associativity, Commutativity, Unit, Distributivity, Reciprocals and Nonnegative Products §nontrivial, and every $x\in\mathbb{R}$ with $x\neq0_{\mathbb{R}}$ has some $w\in\mathbb{R}$ with $x\cdot w=1_{\mathbb{R}}$ by Multiplication of Dedekind Cuts: Associativity, Commutativity, Unit, Distributivity, Reciprocals and Nonnegative Products §reciprocal. So $\mathbb{R}$ is a field.

Ordered field. If $x\le y$, then $x+z\le y+z$ by Addition and Order of Dedekind Cuts: an Ordered Abelian Group in Which Every Nonempty Set Bounded Above Has a Supremum §order; if $0_{\mathbb{R}}\le x$ and $0_{\mathbb{R}}\le y$, then $0_{\mathbb{R}}\le x\cdot y$ by Multiplication of Dedekind Cuts: Associativity, Commutativity, Unit, Distributivity, Reciprocals and Nonnegative Products §nonnegative. So $\mathbb{R}$, with $+$, $\cdot$, $0_{\mathbb{R}}$, $1_{\mathbb{R}}$ and $\le$, is an ordered ring, and, being a field, an ordered field.

Clause supremum. Let $s$ be a subset of $\mathbb{R}$ other than the empty set that is bounded above. By Addition and Order of Dedekind Cuts: an Ordered Abelian Group in Which Every Nonempty Set Bounded Above Has a Supremum §complete, $\bigcup s\in\mathbb{R}$ is the supremum of $s$, so $s$ has a supremum.

Clause rationals. By Construction of the Dedekind Cuts: They Form a Set Totally Ordered by Inclusion, Closed under Sums, Negatives and Products of Nonnegative Cuts §rational, $u^{*}\in\mathbb{R}$ for every $u\in\mathbb{Q}$; $\mathbb{Q}$ is a set (The Rational Numbers §rationals) and so is $\mathbb{R}$. Hence, by Maps and Relations Given by Formulas §map, there is a map $\varphi:\mathbb{Q}\to\mathbb{R}$ with $\varphi(u)=u^{*}$ for every $u\in\mathbb{Q}$. It satisfies $\varphi(1)=1^{*}=1_{\mathbb{R}}$ by The Real Numbers §constants, and, for all $u,v\in\mathbb{Q}$, $\varphi(u+v)=(u+v)^{*}=u^{*}+v^{*}=\varphi(u)+\varphi(v)$ by Addition and Order of Dedekind Cuts: an Ordered Abelian Group in Which Every Nonempty Set Bounded Above Has a Supremum §rational-sum and $\varphi(u\cdot v)=(u\cdot v)^{*}=u^{*}\cdot v^{*}=\varphi(u)\cdot\varphi(v)$ by Multiplication of Dedekind Cuts: Associativity, Commutativity, Unit, Distributivity, Reciprocals and Nonnegative Products §rational (sums and products on the left in $\mathbb{Q}$, on the right in $\mathbb{R}$, as The Natural Numbers and the Natural Numbers with Zero: Arithmetic, Order, Induction and Recursion §overloading allows). By clause ordered-field, The Rational Numbers Embed in Exactly One Way into Every Ordered Field §unique applies to $\mathbb{R}$: there is exactly one map $\mathbb{Q}\to\mathbb{R}$ preserving $1$, sums and products. The canonical embedding $\kappa_{\mathbb{R}}$ is that map by The Canonical Embedding of the Rational Numbers into an Ordered Field §embedding, and $\varphi$ is such a map, so $\kappa_{\mathbb{R}}=\varphi$, and $\kappa_{\mathbb{R}}(u)=u^{*}$ for every $u\in\mathbb{Q}$.