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.


General Case

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


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

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


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