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
,
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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
,
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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
,
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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
,
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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
,
,
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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
,
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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) |
Cauchy criteria. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cauchy_criteria&oldid=15239