Real Number Line is Metric Space

Theorem
Let $\R$ be the set of all real numbers.

Let $d: \R \times \R \to \R$ 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 $\R$ and so $\left({\R, d}\right)$ is a metric space.

Proof
From the definition of absolute value:


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

It is clear that this is the same as the euclidean metric on the real vector space $\R^1$.

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

As the real number line is a vector space, it follows that the real number line is a 1-dimensional Euclidean space.