Topology induced by Scaled Euclidean Metric on Positive Integers is Discrete

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

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

Then the metric topology for $\delta$ is a discrete topology.

Proof
Let $\tau_\delta$ denote the metric topology for $\delta$.

In Scaled Euclidean Metric is Metric it is demonstrated that $\delta$ is indeed a metric on $\Z_{>0}$.

Let $a \in \Z_{>0}$.

Recall the definition of the open $\epsilon$-ball of $a$ in $\left({\Z_{>0}, \delta}\right)$:
 * $B_\epsilon \left({a}\right) := \left\{{x \in A: \delta \left({x, a}\right) < \epsilon}\right\}$

Let $x \in \R_{>0}$.

Let $\epsilon \in \R_{>0}$ such that $\epsilon < \dfrac 1 {a \left({a + 1}\right)}$.

But we have:

and so:
 * $\forall x \in \Z_{>0}, x \ne a: x \notin B_\epsilon \left({a}\right)$

It follows that:
 * $B_\epsilon \left({a}\right) := \left\{{a}\right\}$

Thus by definition of $\tau_d$:
 * $\forall a \in \Z_{>0}: \left\{{a}\right\} \in \tau_\delta$

It follows from Basis for Discrete Topology that $\left({\Z_{>0}, \tau_\delta}\right)$ is a discrete topological space.