Complete Metric Spaces
A metric space M is complete if every Cauchy sequenceconverges in M:, so thatasimplies there iswithalso as
Every Euclidean space is complete, as is every closed subset of a complete space. The set of rational numbersusing the absolute value metricare not complete, since a Cauchy sequence inmay 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.
Ifis a complete subset of the metric spacethenis closed inin fact metric space is complete iff it is closed in any containing metric space.
Every complete metric space is a Baire space.
Consider the sequence
Hence givenchoosesuch thatIfthen
Sinceis not convergent inwith the Euclidean metric, it is not convergent in sois a non convergent Cauchy sequence inso not every Cauchy sequence converges insois not complete.
Example: Letandthemis not complete. The sequenceis Cauchy and does not converge. in
Example: Letbe a prime number. On the set of integers, we define
whereThe sequencegiven byis a non convergent Cauchy sequence