# Real Sequence is Cauchy iff Convergent

## Theorem

Let $\left \langle {a_n} \right \rangle$ be a sequence in $\R$.

Then $\left \langle {a_n} \right \rangle$ 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.

$\blacksquare$