# Definition:Cauchy Sequence/Real Numbers

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## Contents

## Definition

Let $\sequence {x_n}$ be a sequence in $\R$.

Then $\sequence {x_n}$ is a **Cauchy sequence** if and only if:

- $\forall \epsilon \in \R_{>0}: \exists N \in \N: \forall m, n \in \N: m, n \ge N: \size {x_n - x_m} < \epsilon$

Considering the real number line as a metric space, it is clear that this is a special case of the definition for a metric space.

## Also see

- Definition:Complete Metric Space: a metric space in which the converse holds, that is a Cauchy sequence is convergent.

Thus in $\R$ a **Cauchy sequence** and a convergent sequence are equivalent concepts.

## Source of Name

This entry was named for Augustin Louis Cauchy.

## Sources

- 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $\S 1.2$: Real Sequences: Definition $1.2.8$ - 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 5$: Subsequences: $\S 5.16$: Cauchy sequences