# Definition:Cauchy Sequence/Cauchy Criterion

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## Definition

The **Cauchy criterion** is the condition:

- For any (strictly) positive real number $\epsilon \in \R_{>0}$, for a sufficiently large natural number $N \in \N$, the difference between the $m$th and $n$th terms of a Cauchy sequence, where $m, n \ge N$, will be less than $\epsilon$.

Informally:

- For any number you care to pick (however small), if you go out far enough into the sequence, past a certain point, the difference between any two terms in the sequence is less than the number you picked.

Or to put it another way, the terms get arbitrarily close together the farther out you go.

## Also known as

Some sources refer to the **Cauchy criterion** as the **Cauchy condition**, but the same term is used for the Cauchy boundary condition.

Hence in order to reduce possible confusion, $\mathsf{Pr} \infty \mathsf{fWiki}$ prefers the use of **Cauchy criterion**

## Source of Name

This entry was named for Augustin Louis Cauchy.

## Sources

- 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next): Entry:**Cauchy condition**or**Cauchy criterion**:**1.**