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.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Cauchy convergence condition: 1.
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Cauchy convergence condition: 1.