The Rational Numbers are Dense in the Real Numbers

Let $\mathbb{R}$ be the real numbers with the order $\le$ of its ordered field structure, let $\mathbb{Q}$ be the set of rational numbers, and let $|s|$ denote the absolute value of $s\in\mathbb{R}$. For $s,t\in\mathbb{R}$ write $s<t$ to mean $s\le t$ and $s\ne t$. Then the following hold.

1. (Between any two reals.) For all $x,y\in\mathbb{R}$ with $x<y$ there is $q\in\mathbb{Q}$ with $x<q<y$.

2. (Approximation.) For every $x\in\mathbb{R}$ and every $\varepsilon\in\mathbb{R}$ with $\varepsilon>0$ there is $q\in\mathbb{Q}$ with $|x-q|<\varepsilon$.

3. (Density.) Let $d$ be the Euclidean distance on $\mathbb{R}$, which makes $\mathbb{R}$ a metric space and which satisfies $d(s,t)=|s-t|$ by the identification of the Euclidean distance on the real line with the absolute value metric, and let $\mathcal{T}_{d}$ be the collection of metric open subsets of $\mathbb{R}$, a topology by the theorem that metric open sets form a topology. Then $\mathbb{Q}$ is a dense subset of the topological space $(\mathbb{R},\mathcal{T}_{d})$.