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:
 * $\forall x, y \in \Z_{>0}: \delta \left({x, y}\right) = \dfrac {\left\lvert{x - y}\right\rvert} {x y}$

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

is a Cauchy sequence in $\left({\Z_{>0}, \delta}\right)$.

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

Then by the Archimedean Principle:
 * $\exists N \in \N: N > \dfrac 1 \epsilon$

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

Thus it can be seen that:
 * $\forall m, n \in \N: m, n > N \implies \delta \left({x_m, x_n}\right) = \left\lvert{\dfrac 1 m - \dfrac 1 n}\right\rvert < \max \left({\dfrac 1 m, \dfrac 1 n}\right) < \dfrac 1 N < \epsilon$

Thus $\left\langle{x_n}\right\rangle$ is a Cauchy sequence in $\left({\Z_{>0}, \delta}\right)$.