Definition:Cauchy Sequence

A rational sequence $$\left \langle {s_n} \right \rangle$$ is a Cauchy sequence if:

$$\forall \epsilon \in \mathbb{Q}, \epsilon > 0: \exists N \in \mathbb{N}: \forall m, n \in \mathbb{N}: m, n \ge N: \left|{s_n - s_m}\right| < \epsilon$$

That is, for any rational number you care to pick (however small), if you go out far enough into the sequence, past a certain point, the difference between any two terms in the sequence is less than the number you picked.