Cauchy criteria

The Cauchy criterion on the convergence of a sequence of numbers: A sequence of (real or complex) numbers , converges to a limit if and only if, given , there exists an such that, for all and ,

The Cauchy criterion on the convergence of a sequence of numbers may be generalized to a criterion on the convergence of points in a complete metric space.

A sequence of points of a complete metric space is convergent if and only if, given , there exists an such that, for all and , the inequality holds.

The Cauchy criterion on the existence of a limit of a function of variables . Let be a function defined on a set in an -dimensional space , taking real or complex values, and let be a limit point of the set (or the symbol , in which case the set is unbounded). There exists a finite limit if and only if, given , there exists a neighbourhood of such that, for any and ,

This criterion may be generalized to more general mappings: Let be a topological space, a limit point of at which the first axiom of countability is valid, a complete metric space, and a mapping of into . Then the limit

exists if and only if, given , there exists a neighbourhood of such that, for all and ,

The Cauchy criterion on the uniform convergence of a family of functions. Let be a set, a topological space with a limit point at which the first axiom of countability holds, a complete metric space, a mapping of the set into , , . Then the family of functions mapping, for a fixed , into is uniformly convergent on as if, given , there exists a neighbourhood of such that, for all , and all ,

In particular, if is the set of natural numbers and , then the sequence is uniformly convergent on as if and only if, given , there exists an such that, for all and all , ,

The Cauchy criterion on the convergence of a series: A series of real numbers is convergent if and only if, given , there exists an such that, for all and all integers ,

The analogue of this criterion for multiple series is known as the Cauchy–Stolz criterion. For example, a double series

is convergent in the sense of convergence of rectangular partial sums

if and only if, given , there exists an such that, for all , and all integers , ,

These criteria generalize to series in Banach spaces (with absolute values replaced by the appropriate norms of the elements).

The Cauchy criterion on the uniform convergence of a series: Let , be functions defined on some set and taking real values. The series

is uniformly convergent on if and only if, given , there exists an such that, for all , all integers and all ,

This criterion also carries over to multiple series, and moreover not only with numerical terms but also with terms in Banach spaces, i.e. to series in which the are mappings of into a Banach space.

The Cauchy criterion on the convergence of improper integrals: Let be a function defined on a half-closed interval , , taking numerical values. Suppose that for any the function is (Riemann- or Lebesgue-) integrable on . Then the improper integral

is convergent if and only if, given , there exists an such that, for all and satisfying the condition , ,

The criterion can be formulated in an analogous way for improper integrals of other types, and it also generalizes to the case in which depends on several variables and assumes values in a Banach space.

The Cauchy criterion on the uniform convergence of improper integrals: Let be some set and suppose that, for every fixed , the function is defined on a half-closed interval , , and takes numerical values. Suppose that for any the function is integrable with respect to on . Then the integral

is uniformly convergent on if and only if, given any , there exists an such that, for any and satisfying the conditions , , and all ,

This criterion also carries over to improper integrals of other types, to functions of several variables and to functions taking values in Banach spaces.

References

[1]	A.L. Cauchy, "Analyse algébrique" , Gauthier-Villars (1821) (German translation: Springer, 1885)
[2]	O. Stolz, Math. Ann. , 24 (1884) pp. 154–171
[3]	J.A. Dieudonné, "Foundations of modern analysis" , Acad. Press (1961) (Translated from French)
[4]	V.A. Il'in, E.G. Poznyak, "Fundamentals of mathematical analysis" , 1–2 , MIR (1971–1973) (Translated from Russian)
[5]	L.D. Kudryavtsev, "A course in mathematical analysis" , 1–2 , Moscow (1981) (In Russian)
[6]	S.M. Nikol'skii, "A course of mathematical analysis" , 1–2 , MIR (1977) (Translated from Russian)
[7]	E.T. Whittaker, G.N. Watson, "A course of modern analysis" , Cambridge Univ. Press (1952) pp. Chapt. 6

Comments

The criterion in 1) can be reformulated as: A sequence of numbers is convergent if and only if it is a Cauchy sequence (see also [a1]). This property of sequences of elements of a metric space is often taken as a definition of completeness of the latter: A metric space is called complete if every Cauchy sequence in it converges.

References

[a1]	W. Rudin, "Principles of mathematical analysis" , McGraw-Hill (1976)