Topologies induced by Usual Metric and Scaled Euclidean Metric on Positive Integers are Homeomorphic

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

Let $d: \Z_{>0} \times \Z_{>0} \to \R$ be the usual (Euclidean) metric on $\Z_{>0}$.

Let $\delta: \Z_{>0} \times \Z_{>0} \to \R$ be the metric on $\Z_{>0}$ defined as:
 * $\forall x, y \in \Z_{>0}: \map \delta {x, y} = \dfrac {\size {x - y} } {x y}$

Let $\tau_d$ denote the metric topology for $d$.

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

Then $\struct {\Z_{>0}, \tau_d}$ and $\struct {\Z_{>0}, \tau_\delta}$ are homeomorphic.

Proof
From Topology induced by Usual Metric on Positive Integers is Discrete‎:
 * $\struct {\Z_{>0}, \tau_d}$ is a discrete space.

From Topology induced by Scaled Euclidean Metric on Positive Integers is Discrete‎:
 * $\struct {\Z_{>0}, \tau_\delta}$ is a discrete space.

Let $I_{\Z_{>0} }$ be the identity mapping from $\Z_{>0}$ to itself.

From Mapping from Discrete Space is Continuous:
 * $I_{\Z_{>0} }: \struct {\Z_{>0}, \tau_d} \to \struct {\Z_{>0}, \tau_\delta}$ is continuous

and:
 * $I_{\Z_{>0} }: \struct {\Z_{>0}, \tau_\delta} \to \struct {\Z_{>0}, \tau_d}$ is continuous.

Hence the result by definition of homeomorphic.