Scaled Euclidean Metric is Metric

Theorem
Let $\R_{>0}$ be the set of (strictly) positive integers.

Let $\delta: \R_{>0} \times \R_{>0} \to \R$ be the metric on $\R_{>0}$ defined as:
 * $\forall x, y \in \R_{>0}: \delta \left({x, y}\right) = \dfrac {\left\lvert{x - y}\right\rvert} {x y}$

Then $\delta$ is a metric.

Proof of $M1$
So axiom $M1$ holds for $\delta$.

Proof of $M2$
So axiom $M2$ holds for $\delta$.

Proof of $M3$
So axiom $M3$ holds for $\delta$.

Proof of $M4$
So axiom $M4$ holds for $\delta$.