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.

$\blacksquare$

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $32$. Special Subsets of the Real Line: $10$