Cauchy Sequence in Positive Integers under Scaled Euclidean Metric

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

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

The sequence $\sequence {x_n}$ in $\Z_{>0}$ defined as:
 * $\forall n \in \N: x_n = n$

is a Cauchy sequence in $\struct {\Z_{>0}, \delta}$.

Proof
For a general $x_m, x_n \in \sequence {x_n}$ as defined:

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

Then by the Axiom of Archimedes:
 * $\exists N \in \N: N > \dfrac 1 \epsilon$

from which it follows that:
 * $\epsilon > \dfrac 1 N$

Thus:

Therefore $\sequence {x_n}$ is a Cauchy sequence in $\struct {\Z_{>0}, \delta}$.