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 iff $$\forall \epsilon \in \mathbb{R}: \epsilon > 0: \exists N: \forall m, n \in \mathbb{N}: m, n \ge N: \left|{x_n - x_m}\right| < \epsilon$$.

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

Cauchy Criterion
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.

See Convergent Sequence is Cauchy Sequence for a proof that a convergent sequence and a Cauchy sequence are (up to equivalence) the same thing.

This condition is known as the Cauchy criterion.