(a) The plot of a Cauchy sequence
shown in blue, as
If the space containing the sequence is complete, the "ultimate destination" of this sequence (that is, the limit) exists.
(b) A sequence that is not Cauchy. The elements
of the sequence fail to get arbitrarily close to each other as the sequence progresses.
In mathematics, a Cauchy sequence is a sequence where the distance between any two elements of the sequence decreases as the sequence progresses. Cauchy sequences are named after Augustin-Louis Cauchy.