Real Number Line is Complete Metric Space

Theorem

The set of real number $\R$, equipped with the usual (Euclidean) metric, forms a complete metric space.

Proof

From Real Number Line is Metric Space, the distance function defined as $d \left({x, y}\right) = \left|{x - y}\right|$ is a metric on $\R$.

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.

$\blacksquare$