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.

$\blacksquare$


Also see