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Cauchy sequence

A Cauchy sequence is a sequence \{x_n\} in a metric space (such as the real numbers with the standard metric) for which the terms become arbitrarily close to one another as the sequence progresses; formally, for every \epsilon > 0, there exists a positive integer N such that |x_m - x_n| < \epsilon whenever m, n > N. This concept, introduced by Augustin-Louis Cauchy in the early 19th century, addresses limitations in defining convergence without presupposing the existence of a limit point, particularly in incomplete spaces like the rational numbers, where sequences may "bunch up" but fail to converge to a rational limit. In the real numbers, which form a complete metric space, every Cauchy sequence converges to some limit within the space—a property known as the Cauchy convergence criterion, equivalent to the completeness axiom for \mathbb{R}. This equivalence underscores the foundational role of Cauchy sequences in real analysis, enabling the rigorous construction of the real numbers from the rationals by adjoining limits of all Cauchy sequences of rationals, thus filling "gaps" in the rationals. Key properties include that every convergent sequence is Cauchy (via the ) and every Cauchy sequence is bounded, though the converse holds only in complete spaces. Cauchy sequences generalize to normed vector spaces and are essential in for defining Banach spaces, where ensures solvability of equations like fixed-point problems. Examples include decimal expansions of real numbers, which form Cauchy sequences converging to the represented value, while the harmonic series partial sums do not qualify as Cauchy due to persistent differences between terms.

Definition

In real numbers

The concept of a Cauchy sequence originated with , who introduced it in his 1821 textbook Cours d'analyse de l'École Royale Polytechnique to provide a rigorous foundation for that does not rely on the prior assumption of a limit point. This approach allowed for the analysis of sequences in terms of their internal consistency, paving the way for modern . In the real numbers \mathbb{R}, a sequence (a_n)_{n=1}^\infty is defined as a Cauchy sequence if, for every \epsilon > 0, there exists a positive integer N such that |a_m - a_n| < \epsilon for all integers m, n > N. This condition ensures that the terms of the sequence become arbitrarily close to each other as n increases, regardless of whether a specific limit is known. Constant sequences, such as a_n = c for all n where c \in \mathbb{R}, are trivially Cauchy, since |a_m - a_n| = 0 < \epsilon for any \epsilon > 0. More generally, every convergent sequence in \mathbb{R} is Cauchy; to see this, suppose \lim_{n \to \infty} a_n = L. Given \epsilon > 0, there exists N such that |a_k - L| < \epsilon/2 for all k > N; thus, for m, n > N, |a_m - a_n| \leq |a_m - L| + |L - a_n| < \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon. For a concrete example, consider the sequence a_n = 1/n, which converges to 0 in \mathbb{R}. For \epsilon > 0, choose N > 2/\epsilon; then for m, n > N, \left| \frac{1}{m} - \frac{1}{n} \right| \leq \frac{1}{N} < \frac{\epsilon}{2} < \epsilon, confirming it is Cauchy. Focusing first on Cauchy sequences in the real numbers establishes foundational intuition, as \mathbb{R} with the absolute value metric serves as the standard introductory framework in real analysis before extending to broader structures.

In metric spaces

In a metric space (X, d), a sequence (x_n)_{n=1}^\infty in X is called a Cauchy sequence if, for every \varepsilon > 0, there exists a positive N such that d(x_m, x_n) < \varepsilon whenever m, n \geq N. This condition captures the intuitive notion that the terms of the sequence eventually become arbitrarily close to each other, regardless of the specific structure of X. The definition in the real numbers is a special case of this general formulation, where X = \mathbb{R} and the metric is the standard absolute value metric d(x, y) = |x - y|. In arbitrary metric spaces, the distance function d allows the concept to apply to diverse settings, such as function spaces equipped with the supremum metric or graphs with shortest-path distances, without relying on the linear order of the reals. A fundamental property of Cauchy sequences in metric spaces is that they are always bounded. To see this, fix \varepsilon = 1; there exists N such that d(x_m, x_n) < 1 for all m, n \geq N. Let r = \max\{d(x_1, x_N), \dots, d(x_{N-1}, x_N), 1\}. Then every x_k satisfies d(x_k, x_N) \leq r: if k < N, by the definition of r; if k \geq N, then d(x_k, x_N) < 1 \leq r. Therefore, for any m, n \in \mathbb{N}, d(x_m, x_n) \leq d(x_m, x_N) + d(x_N, x_n) \leq 2r, so the sequence is bounded. Consider the discrete metric space (\mathbb{N}, d), where d(m, n) = 1 if m \neq n and d(m, m) = 0. In this space, all non-empty subsets are isolated points, and a sequence like (n)_{n=1}^\infty (the natural numbers themselves) satisfies d(x_m, x_n) = 1 for all m \neq n, so it fails the Cauchy condition for \varepsilon = 1/2 and does not converge to any point in \mathbb{N}. In contrast, any eventually constant sequence, such as x_n = 1 for n \geq 10 and arbitrary before, is Cauchy (with N = 10) and converges to that constant value, illustrating how the metric enforces "eventual constancy" for Cauchy behavior.

Convergence and Completeness

Modulus of Cauchy convergence

In a metric space (X, d), the modulus of Cauchy convergence for a Cauchy sequence (x_n)_{n \in \mathbb{N}} is a function \phi: \mathbb{Q}^+ \to \mathbb{N} given explicitly by \phi(\varepsilon) = \min \left\{ N \in \mathbb{N} \;\middle|\; \forall m, n \geq N, \, d(x_m, x_n) < \varepsilon \right\}, which quantifies the point from which the sequence terms remain within \varepsilon of each other. This function witnesses the Cauchy property constructively by providing an explicit bound N depending on \varepsilon > 0. The modulus \phi is non-decreasing: if $0 < \varepsilon < \delta, then \phi(\varepsilon) \geq \phi(\delta), since a stronger uniformity condition requires a larger N. Moreover, \phi(\varepsilon) \to \infty as \varepsilon \to 0^+, ensuring the sequence's "Cauchy-ness" strengthens arbitrarily. In the real numbers \mathbb{R}, \phi relates to the rate of convergence: if (x_n) converges to L \in \mathbb{R} with modulus of convergence c(\varepsilon) (the smallest M such that |x_n - L| < \varepsilon for n \geq M), then \phi(\varepsilon/2) \leq c(\varepsilon/2), linking the two notions quantitatively. To construct \phi explicitly from (x_n), one computes or bounds the distances d(x_m, x_n) for increasing m, n until the condition holds for a given \varepsilon, often requiring a recursive search over natural numbers. For example, consider the partial sums s_n = \sum_{k=0}^{n-1} r^k = \frac{1 - r^n}{1 - r} of the geometric series for fixed $0 < r < 1. Assuming m > n, |s_m - s_n| = r^n \frac{1 - r^{m-n}}{1 - r} < \frac{r^n}{1 - r}. Thus, \phi(\varepsilon) is the smallest N such that \frac{r^N}{1 - r} < \varepsilon, or N = \left\lceil \frac{\log(\varepsilon (1 - r))}{\log r} \right\rceil + 1. In constructive mathematics, the modulus \phi enables effective proofs by supplying computable witnesses, avoiding non-constructive existence arguments reliant on the law of excluded middle or choice principles. This approach defines real numbers as Cauchy sequences equipped with such a modulus, ensuring all operations are algorithmic. The concept traces to Hermann Weyl's 1918 predicative foundation of analysis, where moduli formalized uniform convergence in a manner anticipatory of later uniform space developments.

Complete metric spaces

A metric space (X, d) is said to be complete if every Cauchy sequence in X converges to a point in X. This property ensures that the space contains all its limit points with respect to the metric, providing a robust framework for limits and continuity in analysis. Completeness is a fundamental concept that distinguishes certain metric spaces from others, allowing for the consistent application of sequential convergence without "gaps" in the space. The notion of completeness in metric spaces was formalized by Maurice Fréchet in his 1906 doctoral dissertation, where he introduced abstract metric spaces and explored their properties, including convergence of sequences. A key example of a complete metric space is the real line \mathbb{R} equipped with the standard metric d(x, y) = |x - y|. The completeness of \mathbb{R} follows from its least upper bound property: every nonempty subset of \mathbb{R} that is bounded above has a least upper bound in \mathbb{R}. To see this equivalence, consider a Cauchy sequence \{x_n\} in \mathbb{R}; it is bounded, so the set S = \{x_n : n \in \mathbb{N}\} has a supremum \sup S \in \mathbb{R}, and one can show that x_n \to \sup S. Alternatively, completeness can be proved using the nested interval theorem: for a Cauchy sequence, construct nested closed intervals whose lengths tend to zero, and their intersection is a single point serving as the limit./01%3A_Tools_for_Analysis/1.05%3A_The_Completeness_Axiom_for_the_Real_Numbers) Closed subsets of complete metric spaces are themselves complete; for instance, the interval [0, 1] with the subspace metric from \mathbb{R} is complete, as any Cauchy sequence in [0, 1] converges to a point within it. Similarly, Euclidean space \mathbb{R}^n with the Euclidean metric d(\mathbf{x}, \mathbf{y}) = \sqrt{\sum_{i=1}^n (x_i - y_i)^2} is complete, since it is a finite product of complete spaces \mathbb{R}, and completeness preserves under such products. Another important example is the space C[a, b] of continuous functions on a compact interval [a, b] equipped with the supremum norm \|f\|_\infty = \sup_{x \in [a, b]} |f(x)|; this forms a complete metric space, known as a , where uniform limits of continuous functions remain continuous. Complete metric spaces exhibit significant topological properties, such as being Baire spaces: the intersection of countably many dense open sets is dense, a result known as the . This theorem underscores the "large" nature of complete metric spaces, preventing them from being meager (a countable union of nowhere dense sets). The importance of completeness lies in its role in enabling the construction of limits essential for real and functional analysis; without it, many theorems on convergence, fixed points, and operator theory would fail, as seen in the development of and .

Incomplete metric spaces

A metric space is said to be incomplete if it is not complete, meaning there exists at least one Cauchy sequence in the space that does not converge to any point within the space. A classic example of an incomplete metric space is the set of rational numbers \mathbb{Q} equipped with the standard metric d(x, y) = |x - y|. Consider the sequence \{x_n\} defined by x_1 = 1 and x_{n+1} = \frac{1}{2} \left( x_n + \frac{2}{x_n} \right) for n \geq 1. Each x_n is rational, and the sequence is Cauchy in \mathbb{Q} because the terms get arbitrarily close as n increases, yet it converges to \sqrt{2}, which is irrational and thus not in \mathbb{Q}. Another example is the open interval (0, 1) with the subspace metric induced from \mathbb{R}, namely d(x, y) = |x - y|. The sequence x_n = \frac{1}{n} for n \geq 1 lies in (0, 1) and is Cauchy, since for any \epsilon > 0, choosing N > \frac{2}{\epsilon} ensures |x_n - x_m| < \epsilon for all n, m \geq N. However, it converges to $0, which is not in (0, 1). Every metric space admits a completion: a complete metric space into which the original space can be isometrically embedded, preserving distances. For instance, the completion of \mathbb{Q} with the standard metric is \mathbb{R}. This construction addresses incompleteness by adding limit points of Cauchy sequences. The space of all polynomials on [0, 1], denoted P[0, 1], equipped with the supremum norm \|p\|_\infty = \sup_{x \in [0, 1]} |p(x)|, provides yet another example of incompleteness. By the , the polynomials are dense in the space of continuous functions C[0, 1] under the same norm, meaning any continuous function can be uniformly approximated by polynomials. However, since not all continuous functions are polynomials, P[0, 1] is a proper dense subspace and thus not closed in the complete space C[0, 1], implying it is incomplete. These examples highlight "gaps" in incomplete s, where Cauchy sequences fail to converge internally, underscoring the need for completions to achieve a structure where all such sequences have limits.

Properties

In normed spaces

In a ed linear (X, \|\cdot\|), a sequence (x_n)_{n=1}^\infty in X is called a Cauchy sequence if for every \epsilon > 0, there exists N \in \mathbb{N} such that \|x_m - x_n\| < \epsilon whenever m, n > N. This condition leverages the d(x, y) = \|x - y\| induced by the , adapting the general notion to exploit the vector space structure. A fundamental property of Cauchy sequences in normed spaces is their boundedness: there exists M > 0 such that \|x_n\| \leq M for all n \in \mathbb{N}. This follows from the Cauchy criterion, as the terms eventually lie within a of \epsilon around x_N, and the finite initial segment is bounded. Moreover, the set of Cauchy sequences forms a vector subspace, since linear combinations preserve the Cauchy property: if (x_n) and (y_n) are Cauchy sequences and \alpha, \beta are scalars, then (\alpha x_n + \beta y_n) is also Cauchy, by the and subadditivity of the . Normed spaces where every Cauchy sequence converges in the (to an element of the ) are termed s. A canonical example is the \ell^2 of square-summable real or complex sequences (a_n), equipped with the \|(a_n)\|_2 = \sqrt{\sum_{n=1}^\infty |a_n|^2}, which is complete as a consequence of properties of Hilbert spaces or direct verification using estimates. In contrast, not all normed spaces are complete; for instance, the C[0,1] of continuous real-valued functions on [0,1] with the L^1 \|f\|_1 = \int_0^1 |f(t)| \, dt is incomplete. A standard Cauchy sequence demonstrating this is the piecewise linear functions approximating the that is 0 on [0,1/2) and 1 on [1/2,1], which converges in L^1 to a discontinuous function outside C[0,1]. By comparison, the full L^2[0,1] with the L^2 is a . In finite-dimensional normed spaces, such as \mathbb{R}^k with the Euclidean norm, norm convergence of a sequence (x_n) to x implies coordinate-wise (or "pointwise" with respect to the standard basis) convergence: the i-th components satisfy x_n^{(i)} \to x^{(i)} for each i = 1, \dots, k. This equivalence holds because all norms on finite-dimensional spaces are equivalent, ensuring that norm convergence controls component behavior uniformly.

Equivalent formulations

A sequence (x_n) in a metric space is Cauchy if and only if the oscillation of its tails tends to zero, that is, \lim_{N \to \infty} \sup_{m,n \geq N} d(x_m, x_n) = 0. To see this equivalence, suppose (x_n) satisfies the \epsilon-N definition; for given \epsilon > 0, choose N such that d(x_m, x_n) < \epsilon/2 for m,n \geq N, so the supremum is at most \epsilon. Conversely, if the limit of the suprema is zero, for \epsilon > 0 choose N with \sup_{m,n \geq N} d(x_m, x_n) < \epsilon, which implies the Cauchy condition. In the real numbers, the term "fundamental sequence" was historically used as an equivalent characterization of Cauchy sequences, referring to sequences where terms become arbitrarily close for sufficiently large indices. This terminology appears in early 20th-century texts, such as those by G. H. Hardy, emphasizing the foundational role in constructing the reals via equivalence classes. The filter generated by the tails of a sequence provides another equivalent formulation: the sequence is Cauchy if this filter is a Cauchy filter, meaning that for every entourage V, there exists a tail set A in the filter such that A \times A \subset V. This reformulation reduces the \epsilon-N condition to membership in the Fréchet filter of cofinite sets, where tails correspond to sets containing all but finitely many indices. For generalizations beyond sequences indexed by naturals, a net (x_i)_{i \in I} over a directed set I is Cauchy if for every \epsilon > 0, there exists i_0 \in I such that d(x_i, x_j) < \epsilon whenever i, j \geq i_0 in the directed order. This captures "eventual closeness" in partially ordered index sets, preserving the Cauchy property under the directed structure. In uniform spaces, a sequence is Cauchy if it is uniformly close in the uniformity, precisely: for every entourage V, there exists N such that (x_m, x_n) \in V for all m, n \geq N. This view abstracts the metric condition to entourages, enabling definitions without a distance function while maintaining the core notion of tails lying in symmetric neighborhoods. In constructive mathematics, the Cauchy condition is equivalent to the existence of a modulus of convergence, a function specifying N for each \epsilon. In computer science, Cauchy sequences appear in denotational semantics for concurrent programming languages, where they model the limits of approximating processes in complete metric domains, ensuring unique fixed points for iterative computations.

Generalizations

In uniform spaces

In a uniform space (X, \mathcal{U}), where \mathcal{U} is a uniformity consisting of entourages (symmetric subsets of X \times X satisfying certain axioms), a sequence \{x_n\}_{n \in \mathbb{N}} in X is Cauchy if for every entourage U \in \mathcal{U}, there exists N \in \mathbb{N} such that (x_m, x_n) \in U whenever m, n > N. This definition captures the intuitive notion that terms of the sequence become arbitrarily "close" in the sense defined by the uniformity, generalizing the metric case where entourages correspond to \epsilon-neighborhoods. When the uniform space is metrizable, meaning \mathcal{U} is induced by a d, the Cauchy condition reduces precisely to the standard metric definition: for every \epsilon > 0, there exists N such that d(x_m, x_n) < \epsilon for all m, n > N. For instance, in the product uniformity on \mathbb{R}^\mathbb{N} (the set of all of real numbers, equipped with the uniformity generated by finite products of the standard uniformity on \mathbb{R}), a sequence of elements from \mathbb{R}^\mathbb{N} is Cauchy if and only if its projections onto each coordinate are Cauchy in \mathbb{R}. A uniform space is complete if every Cauchy net (a generalization of sequences indexed by directed sets) converges in X. Unlike metric spaces, where sequences suffice due to first countability, nets are necessary in general uniform spaces to ensure completeness captures convergence properly. The framework of uniform spaces, introduced by André Weil in 1937 to axiomatize notions like uniform continuity and compactness without metrics, also links to totally bounded sets: a uniform space is compact if and only if it is complete and totally bounded (every entourage admits a finite cover). This generalization proves advantageous for non-metrizable cases, such as the indiscrete (trivial) uniformity on a set X with more than one point, where the sole entourage is X \times X; here, every is Cauchy, yet the space is not Hausdorff and holds trivially via of all nets to any point.

In topological vector spaces

In a X, a \{x_n\} is Cauchy if, for every neighborhood U of the , there exists N \in \mathbb{N} such that x_m - x_n \in U for all m, n \geq N. This definition aligns with the uniform structure induced by the topology, where the tails of the become arbitrarily small in the sense that their differences lie in symmetric neighborhoods of zero; moreover, since X is a , scalar multiples of these tails and their finite sums remain controlled within scaled neighborhoods, preserving the Cauchy property. In the special case of a normed , the definition reduces to the standard metric one: \|x_m - x_n\| < \epsilon for every \epsilon > 0 and sufficiently large m, n. In the general non-normable case, the condition uses the family of absorbing neighborhoods of the origin to ensure the tails are "small" topologically, without a single metric. For locally convex topological vector spaces, a key property is sequential : every Cauchy sequence converges in the . This holds if and only if the space is sequentially complete, distinguishing it from mere topological , which requires convergence of all Cauchy nets or filters. A prominent example of incompleteness is the space of test functions \mathcal{D}(\mathbb{R}), equipped with its inductive limit topology as a strict inductive limit of Fréchet spaces \mathcal{D}_K(\mathbb{R}) over compact sets K \subset \mathbb{R}. While \mathcal{D}(\mathbb{R}) is sequentially complete, it is not complete, as there exist Cauchy nets that fail to converge within the space. In contrast, many important topological vector spaces are complete. The \mathcal{S}(\mathbb{R}) of rapidly decreasing smooth functions, defined by the countable family of seminorms p_{\alpha,\beta}(f) = \sup_{x \in \mathbb{R}} |x^\alpha D^\beta f(x)|, is a complete Fréchet space. However, spaces of distributions, such as the tempered distributions \mathcal{S}'(\mathbb{R}) or general distributions \mathcal{D}'(\mathbb{R}), arise as duals or completions of these test function spaces to handle non-convergent Cauchy elements in applications like partial differential equations. Barrelled topological spaces, where every closed balanced absorbing set (barrel) is a neighborhood of the origin, play a crucial role in extending properties of Cauchy nets. In such spaces, the holds: a bounded family of continuous linear functionals is equicontinuous, which ensures that Cauchy nets in the space or its behave uniformly, preventing unbounded growth in tails and facilitating results beyond sequences.

In categories

In , the Cauchy completion of a small category \mathcal{C} is constructed as the full subcategory of the presheaf category [\mathcal{C}^\mathrm{op}, \mathbf{Set}] consisting of those presheaves that are retracts of representable functors, equivalently, the closure of \mathcal{C} under absolute colimits; this yields a dense \mathcal{C} \hookrightarrow \hat{\mathcal{C}} where \hat{\mathcal{C}} is Cauchy complete, meaning every idempotent splits. In enriched settings, such as categories enriched over a \mathcal{V} (e.g., metric-enriched categories over \mathbf{Met}), the Cauchy completion functor similarly embeds objects into a completion via colimits of Cauchy filters or absolute weighted colimits, preserving the enrichment. Examples include the Cauchy completion of posets, which for an Artinian poset (a poset where every descending chain stabilizes) is itself already complete, embedding densely into a larger ; similarly, the simplex category \Delta of finite ordinals and order-preserving maps, underlying simplicial sets, is Cauchy complete as its idempotents split naturally. Lawvere metric spaces generalize classical metrics as categories enriched over ([0, \infty], \geq, +), where hom-objects serve as distances; their Cauchy completion recovers the usual metric completion by symmetrizing and quotienting Cauchy sequences in this enriched sense. Freyd's abstract completion extends this to categories with zero morphisms, constructing an exact completion via a reflective subcategory where kernels and cokernels are added universally, applicable to preadditive categories seeking abelian-like structure. In modern , Cauchy sequences formalize synthetic analysis by defining real numbers as quotients of Cauchy sequences of , enabling univalent completions that align with higher-dimensional without classical axioms like . A key theorem states that every small admits a unique up to , as the universal idempotent-complete receiving a dense from the original.