Definition:Cauchy Sequence

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

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

Rational Numbers
The concept can also be defined for rational numbers.

Let $$\left \langle {x_n} \right \rangle$$ be a rational sequence.

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

Comment
That is, for any 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.

Or to put it another way, the terms get closer and closer together the farther out you go.