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.