Cauchy's Convergence Criterion

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

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


Necessary Condition

Suppose $\sequence {x_n}$ is convergent.

From Convergent Sequence in Metric Space is Cauchy Sequence, we have that every convergent sequence in a metric space is a Cauchy sequence.

We also have that the real number line is a metric space.

Hence $\sequence {x_n}$ is a Cauchy sequence.


Sufficient Condition

Suppose $\sequence {x_n}$ is a Cauchy sequence.

We have the result that a Cauchy Sequence Converges on Real Number Line.

Hence $\sequence {x_n}$ is convergent.


The conditions have been shown to be equivalent.

Hence the result.


Also known as

This result is also known as the Cauchy convergence condition.

Source of Name

This entry was named for Augustin Louis Cauchy.