Cauchy's Convergence Criterion

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.

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

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

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

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

$\forall \epsilon \in \R_{>0}: \exists N \in \N: \forall r \in \N: r \ge N: \forall k > 0: \size {\ds \sum_{i \mathop = 1}^k x_{r + i} } < \epsilon$

It can also be styled as the Cauchy convergence criterion.

This entry was named for Augustin Louis Cauchy.

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.
2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Cauchy convergence criterion