Sequence is Cauchy iff Convergent

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Theorem

Real Case

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.


Complex Case

Let $\sequence {z_n}$ be a complex sequence.


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