Let be a metric space. Let be the set of all Cauchy sequences in . Define an equivalence relation on by setting if the interleave sequence of the sequences and is also a Cauchy sequence. The completion of is defined to be the set of equivalence classes of modulo .
The metric on extends to a metric on in the following manner:
where and are representative Cauchy sequences of elements in . The definition of is tailored so that the limit in the above definition is well defined, and the fact that these sequences are Cauchy, together with the fact that is complete, ensures that the limit exists. The space with this metric is of course a complete metric space.
Note the similarity between the construction of and the construction of from . The process used here is the same as that used to construct the real numbers , except for the minor detail that one can not use the terminology of metric spaces in the construction of itself because it is necessary to construct in the first place before one can define metric spaces.
1 Metric spaces with richer structure
If the metric space has an algebraic structure, then in many cases this algebraic structure carries through unchanged to simply by applying it one element at a time to sequences in . We will not attempt to state this principle precisely, but we will mention the following important instances: