## Complete Metric Spaces

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 then is closed in in 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 so is a non convergent Cauchy sequence in so not every Cauchy sequence converges in so is 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 