Cauchy sequence

(a) The plot of a Cauchy sequence [math]\displaystyle{ (x_n), }[/math] shown in blue, as [math]\displaystyle{ x_n }[/math] versus [math]\displaystyle{ n }[/math] 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.