Proof That The Real Numbers With the Euclidean Metric is Complete

Theorem

The real numberswith the Euclidean metricis complete.

Proof

A metric spaceis complete if every Cauchy sequence inconverges to a point in i.e. ifis a sequence inwiththen

Ifis a sequence inthen every element of the sequence is real. If the sequence converges, it must converges to a real number with no complex component so thatis complete with the Euclidean metric.

The open intervalwith the Euclidean metric is not complete because the Cauchy sequence converges to 0 which is not in