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:
 * $\forall x, y \in \Z_{>0}: \map \delta {x, y} = \dfrac {\size {x - y} } {x y}$

Then $\struct {\Z_{>0}, \delta}$ is not a complete metric space.

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

From Cauchy Sequence in Positive Integers under Scaled Euclidean Metric:
 * $\sequence {x_n}$ is a Cauchy sequence in $\struct {\Z_{>0}, \delta}$.

But $\sequence {x_n}$ is not convergent to any $m \in \Z_{>0}$.

Hence the result, by definition of complete metric space.