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 show that this space is complete; i.e. that every Cauchy sequence of real numbers has a limit.

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

Hence the result.