(This limit exists because the real numbers are complete.) This is only a pseudometric, not yet a metric, since two different Cauchy sequences may have the distance 0. But "having distance 0" is an equivalence relation on the set of all Cauchy sequences, and the set of equivalence classes is a metric space, the completion of "M". The original space is embedded in this space via the identification of an element "x" of "M"' with the equivalence class of sequences in "M" converging to "x" (i.e., the equivalence class containing the sequence with constant value "x"). This defines an isometry onto a dense subspace, as required. Notice, however, that this construction makes explicit use of the completeness of the real numbers, so completion of the rational numbers needs a slightly different treatment.