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:
 * $\map d {x_1, x_2} = \size {x_1 - x_2}$

where $\size x$ is the absolute value of $x$.

Then $d$ is a metric on $\R$ and so $\struct {\R, d}$ is a metric space.

Proof of $M1$
So axiom $M1$ holds for $d$.

Proof of $M2$
So axiom $M2$ holds for $d$.

Proof of $M3$
So axiom $M3$ holds for $d$.

Proof of $M4$
So axiom $M4$ holds for $d$.