Positive Integers under Scaled Euclidean Metric is not Complete Metric Space

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

Then $\left({\Z_{>0}, \delta}\right)$ is not a complete metric space.

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

From Cauchy Sequence in Positive Integers under Scaled Euclidean Metric:
 * $\left\langle{x_n}\right\rangle$ is a Cauchy sequence in $\left({\Z_{>0}, \delta}\right)$.

But $\left\langle{x_n}\right\rangle$ is not convergent to any $m \in \Z_{>0}$.

Hence the result, by definition of complete metric space.