Proof of Rational Tuples and Rational Matrices Approximating a Real Tuple or Matrix from Below and Above in Every Component

A general step. Let $L$ be a finite set, $w:L\to\mathbb{R}$ a map, with $w_{l}=w(l)$, and $\delta\in\mathbb{R}$ with $0<\delta$. Then there are maps $f,g:L\to\mathbb{Q}$ with $w_{l}-\delta<\kappa(f(l))<w_{l}<\kappa(g(l))<w_{l}+\delta$ for every $l\in L$. For $l\in L$ let

$$C_{l}=\{u\in\mathbb{Q}:w_{l}-\delta<\kappa(u)<w_{l}\},\qquad D_{l}=\{u\in\mathbb{Q}:w_{l}<\kappa(u)<w_{l}+\delta\},$$

subsets of $\mathbb{Q}$; $l\mapsto C_{l}$ and $l\mapsto D_{l}$ are families of sets indexed by $L$, maps from $L$ to $\mathcal{P}(\mathbb{Q})$ given by these expressions (Maps and Relations Given by Formulas (i)). Let $l\in L$. The field $\mathbb{R}$ is an ordered field in which every nonempty subset that is bounded above has a supremum, and its canonical embedding is $\kappa$, by The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness (vii); so Infima, the Archimedean Property, Density of the Rationals and Rational Approximation from Below in an Ordered Field Whose Nonempty Sets Bounded Above Have Suprema (iii), applied to $w_{l}-\delta<w_{l}$ and to $w_{l}<w_{l}+\delta$, which follow from $0<\delta$ by Rules of Order in an Ordered Ring: Differences, Sums, Negatives, Products and Squares (ii), adding $w_{l}-\delta$, respectively $w_{l}$, to both sides, gives $u\in\mathbb{Q}$ with $w_{l}-\delta<\kappa(u)<w_{l}$ and $u'\in\mathbb{Q}$ with $w_{l}<\kappa(u')<w_{l}+\delta$. Hence $C_{l}\neq\emptyset$ and $D_{l}\neq\emptyset$ for every $l\in L$. As $L$ is finite, Finite Sets: the Pigeonhole Principle, Uniqueness of the Length, Subsets, Unions, Products, Images, Bounded Sets of Natural Numbers, Extreme Elements, Sets of Maps, Finite Unions and Finite Choice (xi) gives a map $f:L\to\bigcup_{l\in L}C_{l}$ with $f(l)\in C_{l}$ for every $l\in L$, and a map $g:L\to\bigcup_{l\in L}D_{l}$ with $g(l)\in D_{l}$ for every $l\in L$. Both unions are subsets of $\mathbb{Q}$, by Indexed Families of Sets and Their Union, Intersection and Product (ii), so $f$ and $g$ are maps $L\to\mathbb{Q}$, and $f(l)\in C_{l}$ and $g(l)\in D_{l}$ are the required inequalities.

Proof of (i). $[n]$ is finite by Finite Sets: the Pigeonhole Principle, Uniqueness of the Length, Subsets, Unions, Products, Images, Bounded Sets of Natural Numbers, Extreme Elements, Sets of Maps, Finite Unions and Finite Choice (vii), since $k\le n$ for every $k\in[n]$ by Intervals of Natural Numbers: Initial Segments, Adding One Element, Splitting and Shifting (i), and $y$ is a map $[n]\to\mathbb{R}$ with $y(i)=y_{i}$, by Euclidean Space $\mathbb{R}^n$: n-Tuples of Real Numbers with Their Componentwise Operations (ii). The general step with $L=[n]$ and $w=y$ gives maps $f,g:[n]\to\mathbb{Q}$, which are elements $a=f$ and $b=g$ of $\mathbb{Q}^{n}$ by Tuples in a Set: the Set of n-Tuples and Their Components (ii), with $a_{i}=f(i)$ and $b_{i}=g(i)$ by Tuples in a Set: the Set of n-Tuples and Their Components (iii); the inequalities of the general step are those of (i).

Proof of (ii). $[p]$ and $[q]$ are finite as in the proof of (i), so $[p]\times[q]$ is finite by Finite Sets: the Pigeonhole Principle, Uniqueness of the Length, Subsets, Unions, Products, Images, Bounded Sets of Natural Numbers, Extreme Elements, Sets of Maps, Finite Unions and Finite Choice (v), and $A$ is a map $[p]\times[q]\to\mathbb{R}$ with $A(i,j)=A_{ij}$, by Real Matrices: the Set of Real m x n Matrices with Their Entrywise Operations (i) and Real Matrices: the Set of Real m x n Matrices with Their Entrywise Operations (ii). The general step with $L=[p]\times[q]$ and $w=A$ gives maps $f,g:[p]\times[q]\to\mathbb{Q}$ with $A_{ij}-\delta<\kappa(f(i,j))<A_{ij}<\kappa(g(i,j))<A_{ij}+\delta$ for all $i\in[p]$ and $j\in[q]$. By Projection, Evaluation and Currying Maps (iii), with $X=[p]$, $I=[q]$ and $Y=\mathbb{Q}$, there is a map $a:[p]\to\mathbb{Q}^{[q]}$ with $a(i)(j)=f(i,j)$ for all $i\in[p]$ and $j\in[q]$, and likewise a map $b:[p]\to\mathbb{Q}^{[q]}$ with $b(i)(j)=g(i,j)$. As $\mathbb{Q}^{[q]}=\mathbb{Q}^{q}$ by Tuples in a Set: the Set of n-Tuples and Their Components (i), $a$ and $b$ are elements of $(\mathbb{Q}^{q})^{p}$ by Tuples in a Set: the Set of n-Tuples and Their Components (ii), and by Tuples in a Set: the Set of n-Tuples and Their Components (iii) their entries are $a_{ij}=a(i)(j)=f(i,j)$ and $b_{ij}=b(i)(j)=g(i,j)$. So the inequalities above are those of (ii).