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

In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, let $\kappa:\mathbb{Q}\to\mathbb{R}$ be the canonical embedding of The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness (iv). For a set $S$ and $m\in\mathbb{N}$, $S^{m}$ is the set of $m$-tuples in $S$, with the components $x_{k}=x(k)$ of $x\in S^{m}$ as in Tuples in a Set: the Set of n-Tuples and Their Components (iii); in particular $\mathbb{R}^{m}$ is Euclidean space. For $p,q\in\mathbb{N}$, $(\mathbb{Q}^{q})^{p}$ is the set of $p$-tuples in $\mathbb{Q}^{q}$, and for $a\in(\mathbb{Q}^{q})^{p}$, $i\in[p]$ and $j\in[q]$, $a_{ij}$ is the $j$-th component of the $i$-th component $a_{i}$ of $a$; $\mathcal{M}_{p\times q}(\mathbb{R})$ is the set of real $p\times q$ matrices, with the entries $A_{ij}$.

(i) Let $n\in\mathbb{N}$, $y\in\mathbb{R}^{n}$ and $\delta\in\mathbb{R}$ with $0<\delta$. There are $a,b\in\mathbb{Q}^{n}$ with

$$y_{i}-\delta<\kappa(a_{i})<y_{i}<\kappa(b_{i})<y_{i}+\delta\quad\text{for every }i\in[n].$$

(ii) Let $p,q\in\mathbb{N}$, $A\in\mathcal{M}_{p\times q}(\mathbb{R})$ and $\delta\in\mathbb{R}$ with $0<\delta$. There are $a,b\in(\mathbb{Q}^{q})^{p}$ with

$$A_{ij}-\delta<\kappa(a_{ij})<A_{ij}<\kappa(b_{ij})<A_{ij}+\delta\quad\text{for all }i\in[p]\text{ and }j\in[q].$$