A metric space M is complete if every Cauchy sequence
converges in M:, so that
as
implies there is
with
also as
Every Euclidean space is complete, as is every closed subset of a complete space. The set of rational numbers
using the absolute value metric
are not complete, since a Cauchy sequence in
may converge to some non rational number.
Every metric space has a unique (up to isometry) completion, which is a complete space that contains the given space as a dense subset. The real numbers are the completion of the rationals.
If
is a complete subset of the metric space
thenis closed inin fact metric space is complete iff it is closed in any containing metric space.
Every complete metric space is a Baire space.
Example: Let
Consider the sequence
Hence given
choose
such that
If
then
Since
is not convergent in
with the Euclidean metric, it is not convergent in sois a non convergent Cauchy sequence inso not every Cauchy sequence converges in
sois not complete.
Example: Let
and
them
is not complete. The sequence
is Cauchy and does not converge. in
Example: Let
be a prime number. On the set of integers, we define
where
The sequence
given by
is a non convergent Cauchy sequence