Definitions for "Cauchy sequence"
A sequence x0, x1, ... of elements of a metric space is said to be a Cauchy sequence if differences |xn+m-xn| are uniformly small in m (i.e. do not depend on m) and tend to 0 as n grows.
sequence {''x''''n''} in a metric space (''M'', ''d'') is a Cauchy sequence if, for every positive real number ''r'', there is an integer ''N'' such that for all integers ''m'', ''n'' ''N'', we have ''d''(''x''''m'', ''x''''n'') ''r''.
a sequence the elements of which get arbitrary close to each other