Real Number Line is Complete Metric Space

Theorem
The set of real numbers $\R$, equipped with the usual Euclidean metric, forms a complete metric space.

Proof
See Real Number Line is Metric Space for the proof that $d \left({x, y}\right) = \left|{x - y}\right|$ is a metric.

It remains to be shown that the metric space $\left({\R, d}\right)$ is complete.

By definition, this is done by demonstrating that every Cauchy sequence of real numbers has a limit.

This is demonstrated in Cauchy Sequence Converges on Real Number Line.

Hence the result.