Complete metric space

Formally, \((x_n)\) is Cauchy when for every \(\varepsilon>0\) there is \(N\) such that \(d(x_m,x_n)<\varepsilon\) for all \(m,n\ge N\). Unlike convergence, this condition mentions no limit: it only compares the terms with one another. That is why it can tell, from the inside, whether a sequence ought to converge.

Every convergent sequence is Cauchy, by the triangle inequality, so completeness asks for the converse. The real numbers are the basic example: it is the supremum axiom phrased with sequences. By contrast \(\mathbb{Q}\) is not complete, because the decimal approximations of \(\sqrt2\) are Cauchy and their limit is irrational; neither is \((0,1]\), where \(1/n\) piles up at a \(0\) that does not belong to the space.

Completeness is a property of the distance and not only of the open sets: \(\mathbb{R}\) with \(|x-y|\) and with \(|\arctan x-\arctan y|\) has the same open sets, but only the first is complete. A typical mistake is to think that «bounded» is enough or that completeness depends only on the shape of the set; what matters is that limits of Cauchy sequences do not escape.

Completeness yields powerful results, such as Banach's fixed point theorem and the Baire category theorem. In the next topic we move from distances to the more general notion of a topology.