# Cauchy's Convergence Criterion

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## Theorem

### Cauchy's Convergence Criterion on Real Numbers

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.

### Cauchy's Convergence Criterion on Complex Numbers

Let $\sequence {z_n}$ be a complex sequence.

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

## Also known as

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

## Source of Name

This entry was named for Augustin Louis Cauchy.

## Sources

- 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.**