Cauchy sequence

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.

(a) The plot of a Cauchy sequence ${\displaystyle (x_{n}),}$ shown in blue, as ${\displaystyle x_{n}}$ versus ${\displaystyle n}$ 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.