Definition:Cauchy Sequence

Metric Space
Let $$M = \left({A, d}\right)$$ be a metric space.

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

Then $$\left \langle {x_n} \right \rangle$$ is a Cauchy sequence iff $$\forall \epsilon \in \R: \epsilon > 0: \exists N: \forall m, n \in \N: m, n \ge N: d \left({x_n, x_m}\right) < \epsilon$$.

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

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

Considering the real number line as a metric space, it is clear that this is a special case of the definition for a metric space.

Rational Numbers
The concept can also be defined for the set of rational numbers $$\Q$$.

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

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

Considering the set of rational numbers as a metric space, it is clear that this is a special case of the definition for a metric space.

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.