Cauchy's Convergence Criterion

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Theorem

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.


Proof

Necessary Condition

Suppose $\sequence {x_n}$ is convergent.

From Convergent Sequence 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.

$\Box$


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.

$\Box$


The conditions have been shown to be equivalent.

Hence the result.

$\blacksquare$


Source of Name

This entry was named for Augustin Louis Cauchy.


Sources