Real Number Line is Metric Space

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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.

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, which is shown in Euclidean Metric is a 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.

$\blacksquare$