I'm not sure what definition you have. To me a sequence $\{x_n\}$ is a Cauchy sequence if

$\forall \epsilon > 0~~\exists~ m,n \in \mathbb{N} \ni \left|x_n-x_m\right| < \epsilon$

i.e. the elements get arbitrarily close to one another as the sequence progresses.

is this any clearer?