# 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.