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.

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.

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)