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

is a sequence {an} of real numbers for which for every 0, there exists a positive integer n0 such that | am - an| , whenever m n ³ n0.

A sequence {} in a metric space with metric is called a Cauchy sequence (or Cauchy for short) if for every positive real number , there is an integer such that for all integers and greater than , the distance (, ) is less than .

In mathematics, a Cauchy sequence, named after Augustin Cauchy, is a sequence whose elements become close as the sequence progresses. To be more precise, by dropping a finite number of elements from the start of the sequence we can make the maximum distance between any two remaining elements arbitrarily small.