Cauchy sequence

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(a) The plot of a Cauchy sequence  shown in blue, as  versus  If the space containing the sequence is complete, the "ultimate destination" of this sequence (that is, the limit) exists.
(a) The plot of a Cauchy sequence (x_n), shown in blue, as x_n versus 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.
(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 two consecutive elements of the sequence decreases. Cauchy sequences are named after Augustin-Louis Cauchy.