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The Cauchy criterion on the convergence of a sequence of numbers: A sequence of (real or complex) numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c0208301.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c0208302.png" /> converges to a limit if and only if, given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c0208303.png" />, there exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c0208304.png" /> such that, for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c0208305.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c0208306.png" />,
+
{{MSC|40A05}}
 +
{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c0208307.png" /></td> </tr></table>
+
===The elementary Cauchy criterion for sequences of real numbers===
 +
The Cauchy criterion is a characterization of convergent sequences of real numbers. More precisely it states that
  
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.
+
'''Theorem 1'''
 +
A sequence $\{a_n\}$ of real numbers has a finite limit if and only if for every $\varepsilon > 0$ there is an $N$ such that
 +
\begin{equation}\label{e:cauchy}
 +
|a_n-a_m| < \varepsilon \qquad \mbox{for every}\;\;  n,m \geq N\, .
 +
\end{equation}
  
A sequence of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c0208308.png" /> of a complete metric space is convergent if and only if, given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c0208309.png" />, there exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083010.png" /> such that, for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083012.png" />, the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083013.png" /> holds.
+
The latter is often called the Cauchy condition and a sequence which satisfies it is called [[Cauchy sequence]]. An intuitive way of thinking about a Cauchy sequence is that it oscillates less and less. More precisely we could introduce the oscillation after the $N$-th element as
 +
\[
 +
O (N) := \sup \big\{ |a_n-a_m| : n,m\geq N\big\}\,  
 +
\]
 +
and hence the Cauchy condition is equivalent to $\lim_{N\to\infty} O(N) = 0$. Probably the most interesting part of Theorem 1 is that the Cauchy condition implies the existence of the limit: this is indeed related to the [[Completeness (in topology)| completeness]] of the real line.
  
The Cauchy criterion on the existence of a limit of a function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083014.png" /> variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083015.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083016.png" /> be a function defined on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083017.png" /> in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083018.png" />-dimensional space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083019.png" />, taking real or complex values, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083020.png" /> be a limit point of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083021.png" /> (or the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083022.png" />, in which case the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083023.png" /> is unbounded). There exists a finite limit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083024.png" /> if and only if, given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083025.png" />, there exists a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083026.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083027.png" /> such that, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083028.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083029.png" />,
+
The Cauchy criterion can be generalized to a variety of situations, which can all be loosely summarized as "a vanishing oscillation condition is equivalent to convergence".
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083030.png" /></td> </tr></table>
+
===Generalizations to real valued maps===
 +
====Cauchy criterion for series====
 +
Consider a series $\sum_i a_i$ of real numbers. The convergence of the series is by definition the convergence of the sequence of its partial sums
 +
\[
 +
S_j := \sum_{i=0}^j a_i\, .
 +
\]
 +
Thus a straightforward consequence of Theorem 1 is that $\sum_i a_i$ is a convergent series if and only if the sequence $\{S_i\}$ satisfies the Cauchy condition. There are several other criteria (for testing the convergence of a series) which are named after Cauchy: see [[Cauchy test]].
  
This criterion may be generalized to more general mappings: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083031.png" /> be a topological space, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083032.png" /> a limit point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083033.png" /> at which the first axiom of countability is valid, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083034.png" /> a complete metric space, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083035.png" /> a mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083036.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083037.png" />. Then the limit
+
====Cauchy criterion for real-valued functions====
 +
Consider a function $f: A \to \mathbb R$, where $A$ is a subset of the real numbers. Assume $p$ is an [[Accumulation point|accumulation point]] of $A$ (observe that $p$ does not necessarily belong to $A$). We can then introduce the oscillation around $p$ of $f$ as
 +
\[
 +
{\rm osc}\, (f, p, \varepsilon) := \sup \big\{|f(x)-f(y)|: x,y\in (A\setminus \{p\}) \cap ]p-\varepsilon, p+\varepsilon[\big\}\, .
 +
\]
 +
The Cauchy criterion states that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083038.png" /></td> </tr></table>
+
'''Theorem 2'''
 +
The following limit exists and is finite
 +
\begin{equation}\label{e:limit_cont}
 +
\lim_{x\in A, x\to p} f(x)
 +
\end{equation}
 +
if and only if
 +
\[
 +
\lim_{\varepsilon\downarrow 0}\; {\rm osc}\, (f, p, \varepsilon) = 0\, .
 +
\]
  
exists if and only if, given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083039.png" />, there exists a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083040.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083041.png" /> such that, for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083042.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083043.png" />,
+
Analogous statements for $\lim_{x\to\pm \infty}$ hold as well. Since the existence of the limit \eqref{e:limit_cont} can be characterized in terms of sequences, Theorem 2 can be easily reduced to Theorem 1. Theorem 2 can be generalized to maps on a general topological space. For this reason, given a set $A$ and a map $f:A \to \mathbb R$ we define the oscillation of $f$ in $A$ as
 +
\[
 +
{\rm osc}\, (f, A) := \sup \big\{ |f(x)-f(y)|:x,y\in A \big\}\, .
 +
\]
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083044.png" /></td> </tr></table>
+
'''Theorem 3'''
 +
Let $X$ be a topological space, $A\subset X$, $f: A \to \mathbb R$ and $p$ an accumulation point of $A$. Then the following limit exists and is finite
 +
\[
 +
\lim_{x\in A, x\to p} f(x)\,
 +
\]
 +
if and only if for every $\varepsilon >0$ there is a neighborhood $U$ of $p$ such that
 +
\[
 +
{\rm osc}\, (f, (A\cap U)\setminus \{p\}) < \varepsilon\, .
 +
\]
  
The Cauchy criterion on the uniform convergence of a family of functions. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083045.png" /> be a set, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083046.png" /> a topological space with a limit point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083047.png" /> at which the first axiom of countability holds, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083048.png" /> a complete metric space, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083049.png" /> a mapping of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083050.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083051.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083052.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083053.png" />. Then the family of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083054.png" /> mapping, for a fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083055.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083056.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083057.png" /> is uniformly convergent on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083058.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083059.png" /> if, given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083060.png" />, there exists a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083061.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083062.png" /> such that, for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083063.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083064.png" /> and all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083065.png" />,
+
Observe that Theorem 1 can be considered as a particular case of Theorem 1. In fact, consider set $X = \mathbb N \cup \{\infty\}$ endowed with the topology
 +
\[
 +
\tau = \Big\{\emptyset, X\Big\} \cup \Big\{\big\{i\in \mathbb N: i\geq j\big\}\cup \big\{\infty\big\} : j \in \mathbb N \Big\}\, .
 +
\]
 +
A sequence $\{a_i\}$ can be considered as a map $a: \mathbb N \to \mathbb R$. Then the existence of the limit of the sequence is equivalent to the existence of the limit at $\infty$ of the map $a$ on the (subset $A$ of the) topological space $X$.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083066.png" /></td> </tr></table>
+
====Cauchy criterion for improper integrals====
 +
Let $I= [a,b]$ be an interval of the real line and $f:[a,b]\to \mathbb R$ a function which is Riemann (or Lebesgue) integrable on $[a+\varepsilon, b]$ for every $\varepsilon >0$. The [[Improper integral|improper integral]] of $f$ on $I$ is defined as
 +
\[
 +
\int_a^b f(x)\, dx = \lim_{\varepsilon\downarrow 0} \int_{a+\varepsilon}^b f(x)\, dx\, ,
 +
\]
 +
if the latter limit exists.
 +
Similar definitions can be introduced when the function is integrable over intervals of the form $[a, b-\varepsilon]$ or $[a+\delta, b-\varepsilon]$ and when $a=-\infty$ and/or $b=\infty$ (see [[Improper integral]]). If we introduce
 +
\[
 +
F(\varepsilon) := \int_{a+\varepsilon}^b f(x)\, dx
 +
\]
 +
the improper integral is simply $\lim_{\varepsilon\downarrow 0} F(\varepsilon)$ and its existence can therefore be characterized using Theorem 2. For a thorough statement (and all its variants) see [[Improper integral]].
  
In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083067.png" /> is the set of natural numbers and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083068.png" />, then the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083069.png" /> is uniformly convergent on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083070.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083071.png" /> if and only if, given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083072.png" />, there exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083073.png" /> such that, for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083074.png" /> and all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083075.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083076.png" />,
+
===Euclidean spaces===
 +
All the statements above are valid for sequences and series of vectors in $\mathbb R^n$, for functions taking values in $\mathbb R^n$ and for improper integrals of such functions.  
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083077.png" /></td> </tr></table>
+
===Complete metric spaces and Banach spaces===
 +
If $(X,d)$ is a [[Metric space|metric space]], then a Cauchy sequence on $X$ is a sequence $\{x_i\}\subset X$ such that for any $\varepsilon > 0$ there exists an $N$ such that
 +
\[
 +
d(x_n,x_m) < \varepsilon \qquad \forall n,m \geq N\, .
 +
\]
 +
Similarly, one can define a Cauchy sequence in a [[Norm|normed vector space]] using the induced metric. Any convergent sequence in any metric space is necessarily a Cauchy sequence. However, in general metric space not all Cauchy sequences necessarily converge. Those metric spaces for which any Cauchy sequence has a limit are called [[Complete metric space|complete]] and the corresponding versions of Theorem 3 hold. A complete normed vector space is called a [[Banach space]].
  
The Cauchy criterion on the convergence of a series: A series of real numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083078.png" /> is convergent if and only if, given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083079.png" />, there exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083080.png" /> such that, for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083081.png" /> and all integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083082.png" />,
+
An important property of complete metric spaces is that any closed subset is also complete (with the metric induced by the restriction of the ambient metric).
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083083.png" /></td> </tr></table>
+
====Cauchy criterion for uniform convergence====
 +
If $X$ is a set and $\mathcal{B} (X)$ the space of bounded real-valued functions on it, then $\mathcal{B} (X)$ can be endowed with the uniform distance:
 +
\[
 +
\rho (f, g) :=\sup_{x\in X} |f(x) - g(x)|\, .
 +
\]
 +
$(\mathcal{B} (X), \rho)$ is then a complete metric space and as such we conclude the Cauchy criterion for uniform convergence: a sequence $\{f_k\} \subset \mathcal{B} (X)$ converges uniformly if and only if it is a Cauchy sequence for the distance $\rho$. If $X$ has a topological structure, the space of bounded continuous functions $\mathcal{C}_b (X)$ is a closed subset of $(\mathcal{B} (X), d)$. Therefore we also conclude that a Cauchy sequence of bounded continuous functions converges uniformly to a bounded continuous function. A widely used special case of this theorem is when $X$ is a subset of $\mathbb R^n$ or, in particular, an interval $I\subset \mathbb R$ (cf. [[Uniform convergence]]). Corresponding statements can be easily generalized to series of (bounded or continuous) functions
  
The analogue of this criterion for multiple series is known as the Cauchy–Stolz criterion. For example, a double series
+
===Cauchy filters and uniform spaces===
 +
A generalization of the concept of Cauchy sequence is that of [[Cauchy filter]] in a [[uniform space|Uniform space]], to which a corresponding notion of [[Complete uniform space|completeness]] is attached.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083084.png" /></td> </tr></table>
+
===References===
 
+
{|
is convergent in the sense of convergence of rectangular partial sums
+
|-
 
+
|valign="top"|{{Ref|Ca}}|| A.L. Cauchy,  "Analyse algébrique" , Gauthier-Villars  (1821)  (German translation: Springer, 1885)
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083085.png" /></td> </tr></table>
+
|-
 
+
|valign="top"|{{Ref|Di}}|| J.A. Dieudonné,  "Foundations of modern analysis" , Acad. Press  (1961)  (Translated from French)
if and only if, given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083086.png" />, there exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083087.png" /> such that, for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083088.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083089.png" /> and all integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083090.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083091.png" />,
+
|-
 
+
|valign="top"|{{Ref|IP}}|| V.A. Il'in,  E.G. Poznyak,  "Fundamentals of mathematical analysis" , '''1–2''' , MIR  (1971–1973)  (Translated from Russian)
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083092.png" /></td> </tr></table>
+
|-
 
+
|valign="top"|{{Ref|Ku}}|| L.D. Kudryavtsev,  "A course in mathematical analysis" , '''1–2''' , Moscow  (1981)  (In Russian)
These criteria generalize to series in Banach spaces (with absolute values replaced by the appropriate norms of the elements).
+
|-
 
+
|valign="top"|{{Ref|Ni}}|| S.M. Nikol'skii,  "A course of mathematical analysis" , '''1–2''' , MIR  (1977)  (Translated from Russian)
The Cauchy criterion on the uniform convergence of a series: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083093.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083094.png" /> be functions defined on some set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083095.png" /> and taking real values. The series
+
|-
 
+
|valign="top"|{{Ref|Ru}}||  W. Rudin,  "Principles of mathematical analysis" , McGraw-Hill  (1976)
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083096.png" /></td> </tr></table>
+
|-
 
+
|valign="top"|{{Ref|St}}|| O. Stolz,  ''Math. Ann.'' , '''24'''  (1884) pp. 154–171
is uniformly convergent on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083097.png" /> if and only if, given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083098.png" />, there exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c02083099.png" /> such that, for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c020830100.png" />, all integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c020830101.png" /> and all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c020830102.png" />,
+
|-
 
+
|valign="top"|{{Ref|WW}}|| E.T. Whittaker,  G.N. Watson,  "A course of modern analysis" , Cambridge Univ. Press  (1952)
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c020830103.png" /></td> </tr></table>
+
|-
 
+
|}
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c020830104.png" /> are mappings of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c020830105.png" /> into a Banach space.
 
 
 
The Cauchy criterion on the convergence of improper integrals: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c020830106.png" /> be a function defined on a half-closed interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c020830107.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c020830108.png" />, taking numerical values. Suppose that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c020830109.png" /> the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c020830110.png" /> is (Riemann- or Lebesgue-) integrable on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c020830111.png" />. Then the improper integral
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c020830112.png" /></td> </tr></table>
 
 
 
is convergent if and only if, given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c020830113.png" />, there exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c020830114.png" /> such that, for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c020830115.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c020830116.png" /> satisfying the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c020830117.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c020830118.png" />,
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c020830119.png" /></td> </tr></table>
 
 
 
The criterion can be formulated in an analogous way for improper integrals of other types, and it also generalizes to the case in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c020830120.png" /> depends on several variables and assumes values in a Banach space.
 
 
 
The Cauchy criterion on the uniform convergence of improper integrals: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c020830121.png" /> be some set and suppose that, for every fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c020830122.png" />, the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c020830123.png" /> is defined on a half-closed interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c020830124.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c020830125.png" />, and takes numerical values. Suppose that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c020830126.png" /> the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c020830127.png" /> is integrable with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c020830128.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c020830129.png" />. Then the integral
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c020830130.png" /></td> </tr></table>
 
 
 
is uniformly convergent on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c020830131.png" /> if and only if, given any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c020830132.png" />, there exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c020830133.png" /> such that, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c020830134.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c020830135.png" /> satisfying the conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c020830136.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c020830137.png" />, and all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c020830138.png" />,
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020830/c020830139.png" /></td> </tr></table>
 
 
 
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====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.L. Cauchy,  "Analyse algébrique" , Gauthier-Villars  (1821)  (German translation: Springer, 1885)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  O. Stolz,  ''Math. Ann.'' , '''24'''  (1884)  pp. 154–171</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J.A. Dieudonné,  "Foundations of modern analysis" , Acad. Press  (1961)  (Translated from French)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> V.A. Il'in,  E.G. Poznyak,  "Fundamentals of mathematical analysis" , '''1–2''' , MIR  (1971–1973)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> L.D. Kudryavtsev,  "A course in mathematical analysis" , '''1–2''' , Moscow  (1981)  (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> S.M. Nikol'skii,  "A course of mathematical analysis" , '''1–2''' , MIR  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> E.T. Whittaker,  G.N. Watson,  "A course of modern analysis" , Cambridge Univ. Press  (1952) pp. Chapt. 6</TD></TR></table>
 
 
 
 
 
 
 
====Comments====
 
The criterion in 1) can be reformulated as: A sequence of numbers is convergent if and only if it is a [[Cauchy sequence|Cauchy sequence]] (see also [[#References|[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====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Rudin,  "Principles of mathematical analysis" , McGraw-Hill  (1976)</TD></TR></table>
 

Latest revision as of 18:24, 9 December 2013

2020 Mathematics Subject Classification: Primary: 40A05 [MSN][ZBL]

The elementary Cauchy criterion for sequences of real numbers

The Cauchy criterion is a characterization of convergent sequences of real numbers. More precisely it states that

Theorem 1 A sequence $\{a_n\}$ of real numbers has a finite limit if and only if for every $\varepsilon > 0$ there is an $N$ such that \begin{equation}\label{e:cauchy} |a_n-a_m| < \varepsilon \qquad \mbox{for every}\;\; n,m \geq N\, . \end{equation}

The latter is often called the Cauchy condition and a sequence which satisfies it is called Cauchy sequence. An intuitive way of thinking about a Cauchy sequence is that it oscillates less and less. More precisely we could introduce the oscillation after the $N$-th element as \[ O (N) := \sup \big\{ |a_n-a_m| : n,m\geq N\big\}\, \] and hence the Cauchy condition is equivalent to $\lim_{N\to\infty} O(N) = 0$. Probably the most interesting part of Theorem 1 is that the Cauchy condition implies the existence of the limit: this is indeed related to the completeness of the real line.

The Cauchy criterion can be generalized to a variety of situations, which can all be loosely summarized as "a vanishing oscillation condition is equivalent to convergence".

Generalizations to real valued maps

Cauchy criterion for series

Consider a series $\sum_i a_i$ of real numbers. The convergence of the series is by definition the convergence of the sequence of its partial sums \[ S_j := \sum_{i=0}^j a_i\, . \] Thus a straightforward consequence of Theorem 1 is that $\sum_i a_i$ is a convergent series if and only if the sequence $\{S_i\}$ satisfies the Cauchy condition. There are several other criteria (for testing the convergence of a series) which are named after Cauchy: see Cauchy test.

Cauchy criterion for real-valued functions

Consider a function $f: A \to \mathbb R$, where $A$ is a subset of the real numbers. Assume $p$ is an accumulation point of $A$ (observe that $p$ does not necessarily belong to $A$). We can then introduce the oscillation around $p$ of $f$ as \[ {\rm osc}\, (f, p, \varepsilon) := \sup \big\{|f(x)-f(y)|: x,y\in (A\setminus \{p\}) \cap ]p-\varepsilon, p+\varepsilon[\big\}\, . \] The Cauchy criterion states that

Theorem 2 The following limit exists and is finite \begin{equation}\label{e:limit_cont} \lim_{x\in A, x\to p} f(x) \end{equation} if and only if \[ \lim_{\varepsilon\downarrow 0}\; {\rm osc}\, (f, p, \varepsilon) = 0\, . \]

Analogous statements for $\lim_{x\to\pm \infty}$ hold as well. Since the existence of the limit \eqref{e:limit_cont} can be characterized in terms of sequences, Theorem 2 can be easily reduced to Theorem 1. Theorem 2 can be generalized to maps on a general topological space. For this reason, given a set $A$ and a map $f:A \to \mathbb R$ we define the oscillation of $f$ in $A$ as \[ {\rm osc}\, (f, A) := \sup \big\{ |f(x)-f(y)|:x,y\in A \big\}\, . \]

Theorem 3 Let $X$ be a topological space, $A\subset X$, $f: A \to \mathbb R$ and $p$ an accumulation point of $A$. Then the following limit exists and is finite \[ \lim_{x\in A, x\to p} f(x)\, \] if and only if for every $\varepsilon >0$ there is a neighborhood $U$ of $p$ such that \[ {\rm osc}\, (f, (A\cap U)\setminus \{p\}) < \varepsilon\, . \]

Observe that Theorem 1 can be considered as a particular case of Theorem 1. In fact, consider set $X = \mathbb N \cup \{\infty\}$ endowed with the topology \[ \tau = \Big\{\emptyset, X\Big\} \cup \Big\{\big\{i\in \mathbb N: i\geq j\big\}\cup \big\{\infty\big\} : j \in \mathbb N \Big\}\, . \] A sequence $\{a_i\}$ can be considered as a map $a: \mathbb N \to \mathbb R$. Then the existence of the limit of the sequence is equivalent to the existence of the limit at $\infty$ of the map $a$ on the (subset $A$ of the) topological space $X$.

Cauchy criterion for improper integrals

Let $I= [a,b]$ be an interval of the real line and $f:[a,b]\to \mathbb R$ a function which is Riemann (or Lebesgue) integrable on $[a+\varepsilon, b]$ for every $\varepsilon >0$. The improper integral of $f$ on $I$ is defined as \[ \int_a^b f(x)\, dx = \lim_{\varepsilon\downarrow 0} \int_{a+\varepsilon}^b f(x)\, dx\, , \] if the latter limit exists. Similar definitions can be introduced when the function is integrable over intervals of the form $[a, b-\varepsilon]$ or $[a+\delta, b-\varepsilon]$ and when $a=-\infty$ and/or $b=\infty$ (see Improper integral). If we introduce \[ F(\varepsilon) := \int_{a+\varepsilon}^b f(x)\, dx \] the improper integral is simply $\lim_{\varepsilon\downarrow 0} F(\varepsilon)$ and its existence can therefore be characterized using Theorem 2. For a thorough statement (and all its variants) see Improper integral.

Euclidean spaces

All the statements above are valid for sequences and series of vectors in $\mathbb R^n$, for functions taking values in $\mathbb R^n$ and for improper integrals of such functions.

Complete metric spaces and Banach spaces

If $(X,d)$ is a metric space, then a Cauchy sequence on $X$ is a sequence $\{x_i\}\subset X$ such that for any $\varepsilon > 0$ there exists an $N$ such that \[ d(x_n,x_m) < \varepsilon \qquad \forall n,m \geq N\, . \] Similarly, one can define a Cauchy sequence in a normed vector space using the induced metric. Any convergent sequence in any metric space is necessarily a Cauchy sequence. However, in general metric space not all Cauchy sequences necessarily converge. Those metric spaces for which any Cauchy sequence has a limit are called complete and the corresponding versions of Theorem 3 hold. A complete normed vector space is called a Banach space.

An important property of complete metric spaces is that any closed subset is also complete (with the metric induced by the restriction of the ambient metric).

Cauchy criterion for uniform convergence

If $X$ is a set and $\mathcal{B} (X)$ the space of bounded real-valued functions on it, then $\mathcal{B} (X)$ can be endowed with the uniform distance: \[ \rho (f, g) :=\sup_{x\in X} |f(x) - g(x)|\, . \] $(\mathcal{B} (X), \rho)$ is then a complete metric space and as such we conclude the Cauchy criterion for uniform convergence: a sequence $\{f_k\} \subset \mathcal{B} (X)$ converges uniformly if and only if it is a Cauchy sequence for the distance $\rho$. If $X$ has a topological structure, the space of bounded continuous functions $\mathcal{C}_b (X)$ is a closed subset of $(\mathcal{B} (X), d)$. Therefore we also conclude that a Cauchy sequence of bounded continuous functions converges uniformly to a bounded continuous function. A widely used special case of this theorem is when $X$ is a subset of $\mathbb R^n$ or, in particular, an interval $I\subset \mathbb R$ (cf. Uniform convergence). Corresponding statements can be easily generalized to series of (bounded or continuous) functions

Cauchy filters and uniform spaces

A generalization of the concept of Cauchy sequence is that of Cauchy filter in a Uniform space, to which a corresponding notion of completeness is attached.

References

[Ca] A.L. Cauchy, "Analyse algébrique" , Gauthier-Villars (1821) (German translation: Springer, 1885)
[Di] J.A. Dieudonné, "Foundations of modern analysis" , Acad. Press (1961) (Translated from French)
[IP] V.A. Il'in, E.G. Poznyak, "Fundamentals of mathematical analysis" , 1–2 , MIR (1971–1973) (Translated from Russian)
[Ku] L.D. Kudryavtsev, "A course in mathematical analysis" , 1–2 , Moscow (1981) (In Russian)
[Ni] S.M. Nikol'skii, "A course of mathematical analysis" , 1–2 , MIR (1977) (Translated from Russian)
[Ru] W. Rudin, "Principles of mathematical analysis" , McGraw-Hill (1976)
[St] O. Stolz, Math. Ann. , 24 (1884) pp. 154–171
[WW] E.T. Whittaker, G.N. Watson, "A course of modern analysis" , Cambridge Univ. Press (1952)
How to Cite This Entry:
Cauchy criteria. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cauchy_criteria&oldid=15239
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article