Proof of The Real Line and Euclidean Space Are Separable

Let $\kappa:\mathbb{Q}\to\mathbb{R}$ be the canonical embedding (The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness (iv)).

Step 1 (the real line). By The Absolute Value Metric on the Real Numbers (i), $(\mathbb{R},d_{\mathbb{R}})$ is a metric space, and $d_{\mathbb{R}}(s,u)=|s-u|$ for all $s,u\in\mathbb{R}$ (The Distance on the Real Numbers (i), The Distance on the Real Numbers Is a Map with Values the Absolute Differences (i)). The set $\kappa[\mathbb{Q}]\subseteq\mathbb{R}$ is countable, as the image of the countable set $\mathbb{Q}$ (Countable Sets: a Bijection from ℕ onto ℕ×ℕ, Finite Sets, Subsets, Surjective Images, Images, Products, and the Integers and Rationals (ix), Countable Sets: a Bijection from ℕ onto ℕ×ℕ, Finite Sets, Subsets, Surjective Images, Images, Products, and the Integers and Rationals (vi)). It is dense: let $x\in\mathbb{R}$ and $r>0$. By the density of the rationals (The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness (x), 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)) there is $q\in\mathbb{Q}$ with $x-r<\kappa(q)<x$, so

$$d_{\mathbb{R}}\bigl(x,\kappa(q)\bigr)=\bigl|x-\kappa(q)\bigr|<r ,$$

and $\kappa(q)$ lies in the open ball $B_{d_{\mathbb{R}}}(x,r)$. This proves (i) by Separable Metric Space and Separable Real Normed Space (i).

Step 2 (Euclidean space). $(\mathbb{R}^{k},d_{E})$ is a metric space by $\mathbb{R}^n$ with the Euclidean Distance Is a Complete Metric Space with Coordinatewise Convergence (i). Let

$$D=\{\kappa\circ a:\ a\in\mathbb{Q}^{k}\}\subseteq\mathbb{R}^{k},$$

the image of $\mathbb{Q}^{k}$ under $a\mapsto\kappa\circ a$. The set $\mathbb{Q}^{k}$ is countable by Finite Powers of a Countable Set Are Countable (i), as $\mathbb{Q}$ is, so $D$ is countable by Countable Sets: a Bijection from ℕ onto ℕ×ℕ, Finite Sets, Subsets, Surjective Images, Images, Products, and the Integers and Rationals (vi). It is dense: let $x\in\mathbb{R}^{k}$ and $r>0$, and choose a real $\delta>0$ with $k\delta<r$, for instance $\delta=r/(2k)$. By Rational Tuples and Rational Matrices Approximating a Real Tuple or Matrix from Below and Above in Every Component (i) there is $a\in\mathbb{Q}^{k}$ with $x_{i}-\delta<\kappa(a_{i})<x_{i}$, so $|x_{i}-\kappa(a_{i})|\le\delta$, for every $i\in[k]$. A vector all of whose coordinates have absolute value at most $\delta$ has norm at most $k\delta$ (Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square (iv)); with Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square (ii),

$$d_{E}(x,\kappa\circ a)=\lVert x-\kappa\circ a\rVert\le k\delta<r ,$$

so $\kappa\circ a\in D\cap B_{d_{E}}(x,r)$. Hence (ii) follows from Separable Metric Space and Separable Real Normed Space (i).