Is there a rational number between any two irrationals?

Suppose $i_1$ and $i_2$ are distinct irrational numbers with $i_1 < i_2$. Is it necessarily the case that there is a rational number $r$ in the interval $[i_1, i_2]$? How would you construct such a rational number?

[I posted this only so that the useful answers at https://math.stackexchange.com/questions/414036/rationals-and-irrationals-on-the-real-number-line/414048#414048 could be merged here before that question was deleted.]

Answer

Let $x,y\in\mathbb{R}$, $x\neq y$. Without loss of generality, suppose $x<y$. Then there exists a positive $z$ such that $y-x=z$.

By Archimedes’ axiom, there exists a natural number $n$ such that
$$n > \dfrac{1}{z}$$
$$nz > 1$$
$$ny – nx > 1$$
So there exists an integer $m$ such that
$$nx < m < ny$$
$$x < \frac{m}{n} < y$$
i.e. $m/n$ is a rational number between $x$ and $y$.

Since $x$ and $y$ can be any real numbers, in particular they can be irrationals.

Attribution
Source : Link , Question Author : MJD , Answer Author : joeA

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