Real Sequence is Cauchy iff Convergent

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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.

$\blacksquare$