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 $\left({\Z_{>0}, d}\right)$ is a complete metric space.

Proof
Let $\left\langle{x_n}\right\rangle$ be a Cauchy sequence in $\left({\Z_{>0}, d}\right)$.

From Cauchy Sequence in Positive Integers under Usual Metric is eventually Constant:
 * $\left\langle{x_n}\right\rangle$ is a convergent sequence to some $n \in \Z_{>0}$.

Hence the result by definition of complete metric space.