Real Number Line is Complete Metric Space
From ProofWiki
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.
$\blacksquare$
Sources
- Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{II}: \ 28: \ 1$
