Cauchy criteria

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)