Cauchy criteria

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

## Contents

* 1 The elementary Cauchy criterion for sequences of real numbers
* 2 Generalizations to real valued maps
* 2.1 Cauchy criterion for series
* 2.2 Cauchy criterion for real-valued functions
* 2.3 Cauchy criterion for improper integrals
* 3 Euclidean spaces
* 4 Complete metric spaces and Banach spaces
* 4.1 Cauchy criterion for uniform convergence
* 5 Cauchy filters and uniform spaces
* 6 References

### The elementary Cauchy criterion for sequences of real numbers[edit]

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[edit]

#### Cauchy criterion for series[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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[edit]

[Ca] | A.L. Cauchy, "Analyse algébrique" , Gauthier-Villars (1821) (German translation: Springer, 1885)

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