# Real Sequence is Cauchy iff Convergent

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## Theorem

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

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

## Proof

### Necessary Condition

This is demonstrated in Cauchy Sequence Converges on Real Number Line.

$\Box$

### Sufficient Condition

This is demonstrated in Real Convergent Sequence is Cauchy Sequence.

$\blacksquare$