# Cauchy's Convergence Criterion/Real Numbers

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## Theorem

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

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

## Proof

### Necessary Condition

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

Let $\sequence {x_n}$ be convergent.

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

$\Box$

### Sufficient Condition

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

Let $\sequence {x_n}$ be a Cauchy sequence.

Then $\sequence {x_n}$ is convergent.

$\Box$

The conditions are shown to be equivalent.

Hence the result.

$\blacksquare$

## Also known as

**Cauchy's Convergence Criterion** is also known as the **Cauchy convergence condition**.

## Also see

## Source of Name

This entry was named for Augustin Louis Cauchy.

## Sources

- 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $1$: Review of some real analysis: $\S 1.2$: Real Sequences: Theorem $1.2.9$ - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**Cauchy sequence**