Rational Numbers form Metric Space

Theorem
Let $\Q$ be the set of all rational numbers.

Let $d: \Q \times \Q \to \R$ be defined as:


 * $d \left({x_1, x_2}\right) = \left|{x_1 - x_2}\right|$

where $\left|{x}\right|$ is the absolute value of $x$.

Then $d$ is a metric on $\Q$ and so $\left({\Q, d}\right)$ is a metric space.

Proof
From the definition of absolute value:


 * $\left|{x_1 - x_2}\right| = \sqrt {\left({x_1 - x_2}\right)^2}$

This is the same as the Euclidean metric.

This is shown in Euclidean Metric on Real Vector Space is Metric to be a metric.

From Rational Numbers form Vector Space, it follows that the set of all rational numbers is a 1-dimensional Euclidean space.