Definition:Cauchy Sequence

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

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 arbitrarily close together the farther out you go.

This condition is known as the Cauchy criterion.

Also see

 * A Convergent Sequence is Cauchy Sequence.


 * A complete metric space is defined as being a metric space in which the converse holds, i.e. a Cauchy sequence is convergent.


 * The space $\R$ of real numbers is a complete metric space.

Thus in $\R$ a Cauchy sequence and a convergent sequence are equivalent concepts.