Definition:Cauchy Sequence/Real Numbers

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

Then $\left \langle {x_n} \right \rangle$ is a Cauchy sequence iff:
 * $\forall \epsilon \in \R: \epsilon > 0: \exists N: \forall m, n \in \N: m, n \ge N: \left|{x_n - x_m}\right| < \epsilon$

Considering the real number line as a metric space, it is clear that this is a special case of the definition for a metric space.

Also see

 * A Convergent Sequence is Cauchy Sequence.


 * A complete metric space is defined as being a metric space in which the converse holds, i.e. a Cauchy sequence is convergent.


 * The space $\R$ of real numbers is a complete metric space.

Thus in $\R$ a Cauchy sequence and a convergent sequence are equivalent concepts.