Standard Discrete Metric is Metric

Theorem
The standard discrete metric is a metric.

Proof
Let $d: S \times S \to \R$ denote the standard discrete metric on the underlying set $S$ of some space $\left({S, d}\right)$.

By definition:
 * $\forall x, y \in S: d \left({x, y}\right) = \begin{cases}

0 & : x = y \\ 1 & : x \ne y \end{cases}$

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

Proof of $M2$
Let $x = z$.

Let $x \ne z$.

Either $x \ne y$ or $y \ne z$, or both.

So:

So in either case:
 * $d \left({x, y}\right) + d \left({y, z}\right) \ge d \left({x, z}\right)$

and axiom $M2$ holds for $d$.

Proof of $M3$
Let $x \ne y$.

So axiom $M3$ holds for $d$.

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