CauchyFolge

Cauchyfolge, known in English as a Cauchy sequence, is a fundamental concept in analysis. In a metric space (X, d), a sequence (x_n) is called Cauchy if for every positive ε there exists an index N such that for all m, n ≥ N, the distance d(x_m, x_n) is less than ε. In words, the elements of the sequence become arbitrarily close to each other as the sequence advances. In German mathematical literature, a Cauchy sequence may be referred to as a Cauchyfolge.

In complete metric spaces, such as the real numbers with the usual distance, every Cauchy sequence converges to a limit in the same space. More generally, a space is called complete if every Cauchy sequence in it converges to a point of the space. If a space is not complete, a Cauchy sequence may fail to have a limit within the space, though it would converge in a larger completion.

Examples illustrate the concept. In the real numbers, the sequence x_n = 1/n is Cauchy and converges to 0. In the rational numbers, the sequence of rational approximations to sqrt(2) is Cauchy but does not converge to a rational number, highlighting the non-completeness of Q.

Significance and uses. Cauchy sequences are central to the construction of the real numbers and to the formal notion of completeness in analysis. They provide a robust criterion for convergence that does not rely on an explicit limit, and they underpin many convergence tests and the study of function spaces, including Banach spaces, where completeness guarantees the existence of limits of Cauchy sequences.