Theorem
The real numberswith the Euclidean metric
is complete.
Proof
A metric spaceis complete if every Cauchy sequence in
converges to a point in
i.e. if
is a sequence in
with
then
Ifis a sequence in
then every element of the sequence is real. If the sequence converges, it must converges to a real number with no complex component so that
is 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