Cauchy sequence

In mathematical analysis, 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 distance between any two remaining elements arbitrarily small.

Related Topics:
Mathematical analysis - Augustin Cauchy - Sequence - Distance

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Cauchy sequences require the notion of distance so they can only be defined in a metric space. Generalizations to more abstract uniform spaces exist in the form of Cauchy filter and Cauchy net.

Related Topics:
Metric space - Uniform spaces - Cauchy filter - Cauchy net

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

They are of interest because in a complete space, all such sequences converge to a limit, and one can test for "Cauchiness" without knowing the value of the limit (if it exists), in contrast to

Related Topics:
Complete space - Converge to a limit

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

the definition of convergence.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~