# Cauchy's Convergence Criterion

## 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.