Positive Integers under Usual Metric is Complete Metric Space

Theorem

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

Let $d: \Z_{>0} \times \Z_{>0} \to \R$ be the usual (Euclidean) metric on $\Z_{>0}$.

Then $\struct {\Z_{>0}, d}$ is a complete metric space.

Proof

Let $\sequence {x_n}$ be a Cauchy sequence in $\struct {\Z_{>0}, d}$.

From Cauchy Sequence in Positive Integers under Usual Metric is eventually Constant:

$\sequence {x_n}$ is a convergent sequence to some $n \in \Z_{>0}$.

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: $10$