Density of $\mathbb{Q}$ in $\mathbb{R}$. Constructive Proof.

Question

Is there any constructive proof that between two real numbers there is a rational number (I don't think so, but I don't really know)?

Answer 1

Yes. First prove that for each $x\in\mathbb R$ there is some $n\in\mathbb Z$ such that $n\leq x<n+1$ (for this, consider cases for $x$ and use the well ordering principle and the archimedean property). Such $n$ is usually denoted by $\lfloor x\rfloor.$ Next, proceed as follows.

Let $x,y\in\mathbb R$ be such that $y<x.$ By the archimedean property, there is some $n\in\mathbb N$ such that $1<n(x-y)\iff ny+1<nx.$ Now let $m:=\lfloor ny\rfloor.$ Then $ny<m+1\leq ny+1$ and it follows that $ny<m+1<nx$ and hence $y<\dfrac{m+1}{n}<x.$