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===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 (in topology)| 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|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|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|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]].

===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. 

===References===
{|
|-
|valign="top"|{{Ref|Ca}}|| A.L. Cauchy,   "Analyse algébrique" , Gauthier-Villars  (1821)  (German translation: Springer, 1885)
|-
|valign="top"|{{Ref|Di}}|| J.A. Dieudonné,   "Foundations of modern analysis" , Acad. Press  (1961)  (Translated from French)
|-
|valign="top"|{{Ref|IP}}||  V.A. Il'in,   E.G. Poznyak,   "Fundamentals of mathematical analysis" , '''1–2''' , MIR  (1971–1973)  (Translated from Russian)
|-
|valign="top"|{{Ref|Ku}}||  L.D. Kudryavtsev,   "A course in mathematical analysis" , '''1–2''' , Moscow  (1981)  (In Russian)
|-
|valign="top"|{{Ref|Ni}}||  S.M. Nikol'skii,   "A course of mathematical analysis" , '''1–2''' , MIR  (1977)  (Translated from Russian)
|-
|valign="top"|{{Ref|Ru}}||  W. Rudin,   "Principles of mathematical analysis" , McGraw-Hill  (1976)
|-
|valign="top"|{{Ref|St}}|| O. Stolz,   ''Math. Ann.'' , '''24'''  (1884)  pp. 154–171
|-
|valign="top"|{{Ref|WW}}|| E.T. Whittaker,   G.N. Watson,   "A course of modern analysis" , Cambridge Univ. Press  (1952)
|-
|}