Definition:Cauchy Sequence/Real Numbers

Definition
Let $\sequence {x_n}$ be a sequence in $\R$.

Then $\sequence {x_n}$ is a Cauchy sequence :
 * $\forall \epsilon \in \R_{>0}: \exists N \in \N: \forall m, n \in \N: m, n \ge N: \size {x_n - x_m} < \epsilon$

Also see

 * Convergent Sequence in Metric Space is Cauchy Sequence


 * Definition:Complete Metric Space: a metric space in which the converse holds, that is a Cauchy sequence is convergent.


 * Real Number Line is Complete Metric Space

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


 * Cauchy's Convergence Criterion on Real Numbers, where this is made explicit