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:
 * $\forall x, y \in \R_{>0}: \map \delta {x, y} = \dfrac {\size {x - y} } {x y}$

Then $\delta$ is a metric.

Proof of
So holds for $\delta$.

Proof of
So holds for $\delta$.

Proof of
So holds for $\delta$.

Proof of
So holds for $\delta$.