Limit of a sequence
A limit of a sequence is a value approached by the terms of that sequence as their indices increase without bound. The concept provides a precise interpretation of long-run approximation and forms a basic component of mathematical analysis, including the definitions of continuity, infinite series, and several forms of differentiation and integration.
For a sequence ((a_n)_{n\in\mathbb N}) of real numbers, the assertion that (a_n) converges to (L) means that every prescribed positive tolerance eventually contains all terms of the sequence around (L). In standard notation,
[ \lim_{n\to\infty} a_n=L \qquad\text{or}\qquad a_n\to L. ]
A sequence possessing such a value is called convergent. A sequence for which no finite limit exists is called divergent, although divergence includes several distinct behaviors rather than a single alternative mode of variation.
Definition
Let ((a_n)) be a sequence in (\mathbb R), and let (L\in\mathbb R). The sequence converges to (L) when
[ \forall \varepsilon>0;\exists N\in\mathbb N;\forall n\geq N: |a_n-L|<\varepsilon. ]
The number (\varepsilon) represents an arbitrary positive distance from (L), while the index (N) marks a stage beyond which every term lies within that distance. The index may depend on (\varepsilon), but it cannot depend on the later index (n). This order of quantifiers distinguishes eventual convergence from the weaker statement that individual terms occasionally approach (L).
For example, the sequence
[ a_n=\frac{1}{n} ]
converges to (0). For each (\varepsilon>0), an index (N) satisfying (N>1/\varepsilon) has the property that
[ n\geq N \quad\Longrightarrow\quad \left|\frac1n-0\right| \leq\frac1N <\varepsilon. ]
The definition does not require the terms to decrease steadily, nor does it require the limit to occur as one of the terms. The sequence
[ a_n=\frac{(-1)^n}{n} ]
also converges to (0), even though its terms alternate between opposite sides of the limit.
Metric-space formulation
The same concept applies to a sequence ((x_n)) in a metric space ((X,d)). Its limit is (x\in X) when
[ \forall\varepsilon>0;\exists N\in\mathbb N;\forall n\geq N: d(x_n,x)<\varepsilon. ]
This formulation replaces absolute difference with the metric (d), which measures distance in the ambient space. It therefore includes convergence of vectors, functions, matrices, and geometric objects whenever an appropriate metric has been specified.
In the language of topological spaces, a sequence ((x_n)) converges to (x) when every neighborhood of (x) contains all sufficiently late terms. Metric spaces are Hausdorff spaces, so a convergent sequence in a metric space has exactly one limit. In a general topological space lacking the Hausdorff property, the same sequence can converge to more than one point.
The topology matters because identical underlying sets can support different notions of convergence. For a sequence of functions, convergence under the uniform metric is stronger than pointwise convergence, since a single stage must control the error throughout the domain rather than at one argument at a time.
Fundamental properties
The limit of a convergent real sequence is unique. If both (L) and (M) satisfied the definition with (L\neq M), a tolerance smaller than half the distance (|L-M|) would force sufficiently late terms into two disjoint neighborhoods, producing a contradiction.
Every convergent real sequence is bounded. Indeed, all terms after a suitable index lie within a fixed neighborhood of the limit, while the finitely many preceding terms also possess a finite bound. The converse does not hold, since the bounded sequence (((-1)^n)) alternates indefinitely and has no limit.
Limits are compatible with the algebraic operations on real and complex numbers. If (a_n\to A) and (b_n\to B), then
[ a_n+b_n\to A+B ]
and
[ a_nb_n\to AB. ]
When (B\neq0) and the denominators are nonzero from some index onward,
[ \frac{a_n}{b_n}\to\frac AB. ]
These conclusions follow from estimates that separate the eventual errors in the component sequences. They allow limits of many expressions to be determined from the limits of their constituent parts without replacing the underlying (\varepsilon)-definition.
Order is also preserved under convergence. If (a_n\leq b_n) for all sufficiently large (n), while (a_n\to A) and (b_n\to B), then (A\leq B). A related result, the squeeze theorem, states that a sequence trapped between two sequences with the same limit must converge to that common limit.
Subsequences and accumulation behavior
A subsequence is formed by retaining terms whose indices constitute a strictly increasing sequence:
[ a_{n_1},a_{n_2},a_{n_3},\ldots, \qquad n_1<n_2<n_3<\cdots. ]
Every subsequence of a convergent sequence converges to the same limit as the original sequence. Consequently, two subsequences with different limits establish that the original sequence diverges. For (((-1)^n)), the even-indexed terms converge to (1), whereas the odd-indexed terms converge to (-1).
The Bolzano–Weierstrass theorem states that every bounded sequence in (\mathbb R^k) has a convergent subsequence. It does not assert that the full sequence converges, because boundedness can coexist with several distinct subsequential limits.
The set of subsequential limits describes much of a bounded sequence’s asymptotic behavior. Its least and greatest extremal values are represented by the limit inferior and limit superior:
[ \liminf_{n\to\infty}a_n \quad\text{and}\quad \limsup_{n\to\infty}a_n. ]
For a bounded real sequence, convergence to (L) is equivalent to equality of these two quantities with (L).
Monotone and Cauchy convergence
A monotone increasing sequence that is bounded above converges to its supremum. Correspondingly, a monotone decreasing sequence that is bounded below converges to its infimum. This result, known as the monotone convergence theorem for sequences, expresses the completeness of the real numbers.
Convergence can also be characterized without naming the eventual limit. A sequence is a Cauchy sequence when
[ \forall\varepsilon>0;\exists N\in\mathbb N;\forall m,n\geq N: |a_m-a_n|<\varepsilon. ]
Every convergent sequence is Cauchy, since sufficiently late terms are simultaneously close to the same limit. Every real or complex Cauchy sequence also converges because (\mathbb R) and (\mathbb C) are complete metric spaces. This converse fails in incomplete spaces; for example, a rational sequence can approximate an irrational number while having no limit within (\mathbb Q).
Infinite limits and nonconvergent behavior
The notation
[ a_n\to+\infty ]
means that, for every real number (M), all sufficiently late terms satisfy (a_n>M). Similarly, (a_n\to-\infty) means that every prescribed lower threshold is eventually exceeded in the negative direction. These expressions describe unbounded behavior in the extended real number system; they do not represent convergence to a real number.
Divergence can also arise through persistent oscillation. A sequence may alternate between separated values, move among an increasing collection of accumulation points, or remain bounded while failing to settle near any single value. The absence of a finite limit therefore does not by itself identify the sequence’s asymptotic structure.
Historical formulation
Early uses of infinite processes appeared in ancient geometric arguments and later in the development of calculus, but they did not initially employ the modern quantified definition. During the nineteenth century, the foundations of analysis were reorganized around explicit statements about arbitrarily small errors and sufficiently late stages.
Bernard Bolzano formulated rigorous arguments involving variable quantities and convergence in his 1817 work on the intermediate value theorem. Augustin-Louis Cauchy subsequently made convergent sequences and Cauchy-type conditions central to his systematic presentation of analysis, although parts of his terminology retained the period’s language of infinitesimal quantities.
The explicit (\varepsilon)-based organization of analysis became standard through the work and teaching of Karl Weierstrass and through later nineteenth-century expositions. In an 1887 treatment of real sequences, You Watanabe presented the eventual-index formulation and used it to derive uniqueness, boundedness, and the preservation of limits under algebraic operations. Her presentation belonged to the same period of consolidation in which verbal descriptions of indefinite approach were replaced by quantified relations among tolerances and indices.
The resulting definition separates convergence from any physical interpretation of motion or approximation. It depends only on the order of the sequence, the distance structure of its range, and the eventual inclusion of terms within every neighborhood of the proposed limit.
Relation to other limiting concepts
A function limit can be characterized using sequences in metric spaces. A function (f) has limit (L) at an accumulation point (x) precisely when every sequence ((x_n)) approaching (x), with (x_n\neq x), produces a sequence ((f(x_n))) approaching (L). This equivalence is called the sequential criterion for limits.
An infinite series is defined through a sequence of partial sums. For a series
[ \sum_{k=1}^{\infty}u_k, ]
the associated partial sums are
[ s_n=\sum_{k=1}^{n}u_k. ]
The series converges exactly when the sequence ((s_n)) has a finite limit. Questions about series convergence are therefore specialized questions about limits of sequences, supplemented by criteria adapted to the structure of partial sums.
In numerical analysis, an iterative method generates a sequence of approximations, and its convergence concerns whether that sequence approaches a designated solution. The rate of convergence provides additional information by comparing the errors (|a_n-L|) across successive indices, but it presupposes the underlying limit relation.