Proof of Infima, the Archimedean Property, Density of the Rationals and Rational Approximation from Below in an Ordered Field Whose Nonempty Sets Bounded Above Have Suprema

Preliminaries. By hypothesis, $r$ is an ordered field; so $r$ is a set, $+$ and $\cdot$ are binary operations on it, and $\le$ is a total order on $r$, by Fields §field and Ordered Fields §ordered-field. Hence Bounds, Least and Greatest Elements, Suprema and Infima for a Partial Order and Uniqueness of Least and Greatest Elements, Properties of the Strict Order, and Trichotomy for Total Orders apply with $a=r$, and Rules of Arithmetic and Order in an Ordered Field applies to $r$, with $0_{r}$, $1_{r}$ in place of $0$, $1$. Negatives, differences, reciprocals and quotients in $r$ are as in Negatives, Differences, Reciprocals and Quotients §negative and Negatives, Differences, Reciprocals and Quotients §reciprocal. Rearrangements in $r$ that use only associativity, commutativity, distributivity, $z+0_{r}=z$, $z\cdot1_{r}=z$ (all from Commutative Rings §ring), $z+(-z)=0_{r}$ (Negatives, Differences, Reciprocals and Quotients §negative) and $z\cdot z^{-1}=1_{r}$ for $z\neq0_{r}$ (Negatives, Differences, Reciprocals and Quotients §reciprocal) are called ring rearrangements below.

Mixed transitivity. For $a,b,c\in r$, if $a<b$ and $b\le c$, or $a\le b$ and $b<c$, then $a<c$: by Uniqueness of Least and Greatest Elements, Properties of the Strict Order, and Trichotomy for Total Orders §weak-strict the weak inequality is an equality, which gives $a<c$ at once, or strict, and then Uniqueness of Least and Greatest Elements, Properties of the Strict Order, and Trichotomy for Total Orders §strict-transitive gives $a<c$.

Facts about $\kappa$. By The Canonical Embedding of the Rational Numbers into an Ordered Field §embedding, $\kappa$ is the map of The Rational Numbers Embed in Exactly One Way into Every Ordered Field §unique, so for all $u,v\in\mathbb{Q}$: $\kappa(1)=1_{r}$ and $\kappa(u+v)=\kappa(u)+\kappa(v)$ by The Rational Numbers Embed in Exactly One Way into Every Ordered Field §unique; $u<v$ if and only if $\kappa(u)<\kappa(v)$ by The Rational Numbers Embed in Exactly One Way into Every Ordered Field §order; $\kappa(0)=0_{r}$ by The Rational Numbers Embed in Exactly One Way into Every Ordered Field §zero; $\kappa(u-v)=\kappa(u)-\kappa(v)$ by The Rational Numbers Embed in Exactly One Way into Every Ordered Field §negative; and, if $v\neq0$, $\kappa(u/v)=\kappa(u)/\kappa(v)$ with $\kappa(v)\neq0_{r}$ by The Rational Numbers Embed in Exactly One Way into Every Ordered Field §reciprocal. For $n\in\mathbb{N}$, $\kappa(n)$ means $\kappa(j(\iota(n)))$, by The Integers and the Rational Numbers, with the Natural Numbers and the Integers Identified with Subsets of the Rationals §identification; this is defined since $j(\iota(n))\in\mathbb{Q}$. By The Natural Numbers and the Integers inside the Rational Numbers §naturals, $j(\iota(1))=1_{\mathbb{Q}}$, $j(\iota(n+1))=j(\iota(n))+j(\iota(1))$ and $0_{\mathbb{Q}}<j(\iota(n))$, where $n+1\in\mathbb{N}$ by The Natural Numbers and the Natural Numbers with Zero: Arithmetic, Order, Induction and Recursion §background. With the facts above this gives, for every $n\in\mathbb{N}$:

$$\kappa(1)=1_{r},\qquad\kappa(n+1)=\kappa(n)+1_{r},\qquad0_{r}<\kappa(n),\qquad j(\iota(n))\neq0_{\mathbb{Q}},$$

the last by Uniqueness of Least and Greatest Elements, Properties of the Strict Order, and Trichotomy for Total Orders §strict-characterization applied in $\mathbb{Q}$, whose order is a total order by The Rational Numbers Form an Archimedean Ordered Field Containing the Integers §ordered-field. These are the facts (K).

Clause Infima, the Archimedean Property, Density of the Rationals and Rational Approximation from Below in an Ordered Field Whose Nonempty Sets Bounded Above Have Suprema §infimum. Let $s$ be a nonempty subset of $r$ that is bounded below, and fix a lower bound $b\in r$ of $s$ (Bounds, Least and Greatest Elements, Suprema and Infima for a Partial Order §bounded) and an element $v_{0}\in s$. Let

$$t=\{z\in r:\exists v\,(v\in s\wedge z=-v)\},$$

formed by class abstraction from a formula quantifying over sets only, in which $-v$ is used only for $v\in s\subseteq r$, where it is defined; $t$ is a subclass of the set $r$, hence a set by Subclasses of Sets Are Sets, the Union and Power Set of a Set Exist Uniquely, Binary Unions of Sets Are Sets, and the Universal Class Is Proper §subclass. It is nonempty, since $-v_{0}\in t$. It is bounded above by $-b$: if $z\in t$, say $z=-v$ with $v\in s$, then $b\le v$, so $-v\le-b$ by Rules of Arithmetic and Order in an Ordered Field §order-negative. By the hypothesis on $r$, $t$ has a supremum $m$, the least element of $\operatorname{Ub}(t)$ (Bounds, Least and Greatest Elements, Suprema and Infima for a Partial Order §supremum). We show that $-m$ is the greatest element of $\operatorname{Lb}(s)$, i.e. an infimum of $s$ by Bounds, Least and Greatest Elements, Suprema and Infima for a Partial Order §supremum.

First, $-m\in\operatorname{Lb}(s)$: for $v\in s$ we have $-v\in t$, so $-v\le m$, hence $-m\le-(-v)=v$ by Rules of Arithmetic and Order in an Ordered Field §order-negative and Rules of Arithmetic and Order in an Ordered Field §signs. Second, let $c\in\operatorname{Lb}(s)$. Then $-c\in\operatorname{Ub}(t)$: for $z\in t$, $z=-v$ with $v\in s$, so $c\le v$ and $z=-v\le-c$ by Rules of Arithmetic and Order in an Ordered Field §order-negative. As $m$ is the least element of $\operatorname{Ub}(t)$, $m\le-c$, hence $c=-(-c)\le-m$ by Rules of Arithmetic and Order in an Ordered Field §order-negative and Rules of Arithmetic and Order in an Ordered Field §signs. So $-m$ is the greatest element of $\operatorname{Lb}(s)$.

Clause Infima, the Archimedean Property, Density of the Rationals and Rational Approximation from Below in an Ordered Field Whose Nonempty Sets Bounded Above Have Suprema §archimedean. Suppose, for a contradiction, that $x<\kappa(n)$ fails for every $n\in\mathbb{N}$. Then $\kappa(n)\le x$ for every $n\in\mathbb{N}$ by Uniqueness of Least and Greatest Elements, Properties of the Strict Order, and Trichotomy for Total Orders §total-negation. Let

$$K=\{z\in r:\exists n\,(n\in\mathbb{N}\wedge z=\kappa(n))\},$$

formed from a formula quantifying over sets only, with $\kappa(n)=\kappa(j(\iota(n)))$ defined for $n\in\mathbb{N}$; $K$ is a subclass of $r$, hence a set by Subclasses of Sets Are Sets, the Union and Power Set of a Set Exist Uniquely, Binary Unions of Sets Are Sets, and the Universal Class Is Proper §subclass. It is nonempty, as $\kappa(1)\in K$ ($1\in\mathbb{N}$ by The Natural Numbers and the Natural Numbers with Zero: Arithmetic, Order, Induction and Recursion §background), and $x$ is an upper bound of it. By the hypothesis on $r$ it has a supremum $m$, the least element of $\operatorname{Ub}(K)$.

From $0_{r}<1_{r}$ (Rules of Arithmetic and Order in an Ordered Field §squares), adding $m-1_{r}$ by Rules of Arithmetic and Order in an Ordered Field §order-sum and using ring rearrangements, $m-1_{r}<m$. Hence $m\le m-1_{r}$ fails by Uniqueness of Least and Greatest Elements, Properties of the Strict Order, and Trichotomy for Total Orders §total-negation, so $m-1_{r}\notin\operatorname{Ub}(K)$, since otherwise $m\le m-1_{r}$ as $m$ is least in $\operatorname{Ub}(K)$. So there is $z\in K$ for which $z\le m-1_{r}$ fails; fix one and fix $n\in\mathbb{N}$ with $z=\kappa(n)$. Then $m-1_{r}<\kappa(n)$ by Uniqueness of Least and Greatest Elements, Properties of the Strict Order, and Trichotomy for Total Orders §total-negation, and adding $1_{r}$ by Rules of Arithmetic and Order in an Ordered Field §order-sum, $m<\kappa(n)+1_{r}=\kappa(n+1)$ by (K). But $n+1\in\mathbb{N}$, so $\kappa(n+1)\in K$ and $\kappa(n+1)\le m$, which by Uniqueness of Least and Greatest Elements, Properties of the Strict Order, and Trichotomy for Total Orders §total-negation contradicts $m<\kappa(n+1)$. Hence some $n\in\mathbb{N}$ has $x<\kappa(n)$. Since $x\in r$ was arbitrary, this holds for every element of $r$; it is used below for elements other than $x$.

Clause Infima, the Archimedean Property, Density of the Rationals and Rational Approximation from Below in an Ordered Field Whose Nonempty Sets Bounded Above Have Suprema §density. Let $x<y$. The choices are made in the following order.

Step 1: the gap. Let $d=y-x$. By Rules of Arithmetic and Order in an Ordered Field §order-sum, adding $-x$ to $x<y$ gives $0_{r}<d$. So $d\neq0_{r}$ (Uniqueness of Least and Greatest Elements, Properties of the Strict Order, and Trichotomy for Total Orders §strict-characterization), $d^{-1}$ is defined, and $0_{r}<d^{-1}$ by Rules of Arithmetic and Order in an Ordered Field §positive-reciprocal.

Step 2: the scale. By clause Infima, the Archimedean Property, Density of the Rationals and Rational Approximation from Below in an Ordered Field Whose Nonempty Sets Bounded Above Have Suprema §archimedean applied to $d^{-1}$, fix $m\in\mathbb{N}$ with $d^{-1}<\kappa(m)$, and put $M=\kappa(m)$; $0_{r}<M$ by (K). Multiplying by $d$, with $0_{r}<d$, by Rules of Arithmetic and Order in an Ordered Field §order-product, $1_{r}=d^{-1}\cdot d<M\cdot d$. By ring rearrangements and Rules of Arithmetic and Order in an Ordered Field §signs, $M\cdot d=M\cdot y-M\cdot x$, and adding $M\cdot x$ by Rules of Arithmetic and Order in an Ordered Field §order-sum,

$$M\cdot x+1_{r}<M\cdot y.$$

Step 3: the shift. By clause Infima, the Archimedean Property, Density of the Rationals and Rational Approximation from Below in an Ordered Field Whose Nonempty Sets Bounded Above Have Suprema §archimedean applied to $-(M\cdot x)$, fix $p\in\mathbb{N}$ with $-(M\cdot x)<\kappa(p)$, and put $w=M\cdot x+\kappa(p)$. Adding $M\cdot x$ by Rules of Arithmetic and Order in an Ordered Field §order-sum, $0_{r}<w$.

Step 4: the least natural number above $w$. Let $A=\{k\in\mathbb{N}:w<\kappa(k)\}$, formed from a formula quantifying over sets only; it is a subclass of the set $\mathbb{N}$, hence a set by Subclasses of Sets Are Sets, the Union and Power Set of a Set Exist Uniquely, Binary Unions of Sets Are Sets, and the Universal Class Is Proper §subclass, and nonempty by clause Infima, the Archimedean Property, Density of the Rationals and Rational Approximation from Below in an Ordered Field Whose Nonempty Sets Bounded Above Have Suprema §archimedean applied to $w$. By Arithmetic and Order of the Natural Numbers §well-order, fix $k\in A$ with $k\le v$ for every $v\in A$. Then $w<\kappa(k)$, and we claim $\kappa(k)\le w+1_{r}$. If $k=1$, then $\kappa(k)=1_{r}=0_{r}+1_{r}\le w+1_{r}$ by (K), $0_{r}\le w$ (from $0_{r}<w$ by Uniqueness of Least and Greatest Elements, Properties of the Strict Order, and Trichotomy for Total Orders §weak-strict) and Rules of Arithmetic and Order in an Ordered Field §order-sum (with reflexivity of $\le$ for the second summand). If $k\neq1$, fix $e\in\mathbb{N}$ with $k=e+1$ by Arithmetic and Order of the Natural Numbers §predecessor. Then $e<k$ by Arithmetic and Order of the Natural Numbers §successor, so $k\le e$ fails by Arithmetic and Order of the Natural Numbers §trichotomy and Arithmetic and Order of the Natural Numbers §partial-order; hence $e\notin A$, i.e. $w<\kappa(e)$ fails, so $\kappa(e)\le w$ by Uniqueness of Least and Greatest Elements, Properties of the Strict Order, and Trichotomy for Total Orders §total-negation, and $\kappa(k)=\kappa(e)+1_{r}\le w+1_{r}$ by (K) and Rules of Arithmetic and Order in an Ordered Field §order-sum. In both cases

$$w<\kappa(k)\le w+1_{r}.$$

Step 5: the rational number. Since $j(\iota(m))\neq0_{\mathbb{Q}}$ by (K), let $u=(j(\iota(k))-j(\iota(p)))/j(\iota(m))\in\mathbb{Q}$, i.e. $u=(k-p)/m$ read in $\mathbb{Q}$. By the facts about $\kappa$, $\kappa(u)=(\kappa(k)-\kappa(p))/M$; as $M\neq0_{r}$ (from $0_{r}<M$ by Uniqueness of Least and Greatest Elements, Properties of the Strict Order, and Trichotomy for Total Orders §strict-characterization), ring rearrangements give $\kappa(u)\cdot M=\kappa(k)-\kappa(p)$. Adding $-\kappa(p)$ to the display of Step 4 by Rules of Arithmetic and Order in an Ordered Field §order-sum, and using $w-\kappa(p)=M\cdot x$ (ring rearrangements), we get

$$M\cdot x<\kappa(u)\cdot M\le M\cdot x+1_{r}.$$

With Step 2 and mixed transitivity, $\kappa(u)\cdot M<M\cdot y$. Since $0_{r}<M$, Rules of Arithmetic and Order in an Ordered Field §order-product and commutativity turn $x\cdot M<\kappa(u)\cdot M$ and $\kappa(u)\cdot M<y\cdot M$ into $x<\kappa(u)$ and $\kappa(u)<y$. So $u\in\mathbb{Q}$ satisfies $x<\kappa(u)<y$.

Clause Infima, the Archimedean Property, Density of the Rationals and Rational Approximation from Below in an Ordered Field Whose Nonempty Sets Bounded Above Have Suprema §rational-supremum. Let $L=\{z\in r:\exists u\,(u\in\mathbb{Q}\wedge z=\kappa(u)\wedge z<x)\}$, a subset of $r$ by Subclasses of Sets Are Sets, the Union and Power Set of a Set Exist Uniquely, Binary Unions of Sets Are Sets, and the Universal Class Is Proper §subclass. We show that $x$ is the least element of $\operatorname{Ub}(L)$, i.e. the supremum of $L$ by Bounds, Least and Greatest Elements, Suprema and Infima for a Partial Order §supremum.

$x\in\operatorname{Ub}(L)$: every $z\in L$ has $z<x$, hence $z\le x$ by Uniqueness of Least and Greatest Elements, Properties of the Strict Order, and Trichotomy for Total Orders §weak-strict.

$x$ is least in $\operatorname{Ub}(L)$: let $b\in\operatorname{Ub}(L)$ and suppose $x\le b$ fails. Then $b<x$ by Uniqueness of Least and Greatest Elements, Properties of the Strict Order, and Trichotomy for Total Orders §total-negation, and by clause Infima, the Archimedean Property, Density of the Rationals and Rational Approximation from Below in an Ordered Field Whose Nonempty Sets Bounded Above Have Suprema §density fix $u\in\mathbb{Q}$ with $b<\kappa(u)<x$. Then $\kappa(u)\in L$, so $\kappa(u)\le b$, contradicting $b<\kappa(u)$ by Uniqueness of Least and Greatest Elements, Properties of the Strict Order, and Trichotomy for Total Orders §total-negation. Hence $x\le b$, and $x$ is the supremum of $L$.