# Cauchy's Convergence Criterion

## Contents

## Theorem

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

Then $\sequence {x_n}$ is convergent if and only if $\sequence {x_n}$ is a Cauchy sequence.

## Proof

### Necessary Condition

Suppose $\sequence {x_n}$ is convergent.

From Convergent Sequence is Cauchy Sequence, we have that every convergent sequence in a metric space is a Cauchy sequence.

We also have that the real number line is a metric space.

Hence $\sequence {x_n}$ is a Cauchy sequence.

$\Box$

### Sufficient Condition

Suppose $\sequence {x_n}$ is a Cauchy sequence.

We have the result that a Cauchy Sequence Converges on Real Number Line.

Hence $\sequence {x_n}$ is convergent.

$\Box$

The conditions have been shown to be equivalent.

Hence the result.

$\blacksquare$

## Also known as

This result is also known as the **Cauchy convergence condition**.

## 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: Theorem $1.2.9$ - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**Cauchy convergence condition**:**1.** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**Cauchy convergence condition**:**1.**