Cauchy sequence

Let \(X\) be a metric space. Then a sequence \((x_1,x_2,x_3,\ldots)\) of elements of \(X\) is said to be Cauchy if given any \(\varepsilon > 0\) there is a natural number \(N\) such that the distance between \(x_i\) and \(x_j\) is less than \(\varepsilon\) whenever \(i\) and \(j\) are greater than \(N\). In effect, successive elements of the sequence eventually become arbitrarily close together.

A metric space is said to be complete if and only if every Cauchy sequence is also a convergent sequence.