# Sequence

A sequence is a list of objects in a specified order, most often an infinite list of real numbers written \( (s_1, s_2, s_3, \ldots) \) or \( (s_n) \); formally, a sequence of real numbers is a function from the positive integers to the real numbers<sup>[1](https://mathbooks.unl.edu/Calculus/sec-7-1-sequences.html)</sup>. Sequences are the basic objects through which analysis defines limits, and their convergence theory underlies calculus, the [construction of the real numbers](https://www.edgepedia.org/construction-of-the-real-numbers), iterative numerical algorithms, and the study of series and sequences of functions<sup>[2](https://ocw.mit.edu/courses/18-100a-real-analysis-fall-2020/pages/lecture-notes-and-readings/)</sup>.

| Key fact | Detail |
|---|---|
| Definition | A sequence is a function from the positive integers to a set such as \( \mathbb{R} \); a sequence diverges if it does not converge<sup>[1](https://mathbooks.unl.edu/Calculus/sec-7-1-sequences.html)</sup> |
| Convergence | \( (s_n) \) converges to \( s \) if for every \( \varepsilon > 0 \) there is a real number \( N \) such that \( n > N \) implies \( |s_n - s| < \varepsilon \)<sup>[3](https://math.libretexts.org/Bookshelves/Analysis/Real_Analysis_%28Boman_and_Rogers%29/04%3A_Convergence_of_Sequences_and_Series/4.01%3A_Sequences_of_Real_Numbers)</sup> |
| Subsequence criterion | \( (a_n) \) converges to \( a \) if and only if every subsequence \( (a_{n_k}) \) converges to \( a \)<sup>[4](https://jeanmariedufour.research.mcgill.ca/ResE/Dufour_1992_C_TS_SequencesSeries.pdf)</sup> |
| Limsup–liminf test | A bounded sequence is convergent if and only if \( \limsup (x_n) = \liminf (x_n) \)<sup>[5](https://faculty.ksu.edu.sa/sites/default/files/0471433314realanalysis41.pdf)</sup> |
| Completeness | In the real numbers, a sequence converges exactly when it is Cauchy; Bolzano–Weierstrass states that every bounded sequence has a convergent subsequence<sup>[6](https://neckpinch.com/guide/numbers-limits-and-the-integral/sequences/)</sup> |
| History | Bolzano formulated the modern convergence criterion in 1817, four years before Cauchy's *Cours d'analyse* (1821)<sup>[7](https://plato.stanford.edu/archives/sum2023/entries/bolzano/)</sup> |
| Applications | Iterative numerical methods compute a sequence \( x_0, x_1, x_2, \ldots \) of ever better approximate solutions of \( Ax = b \)<sup>[8](https://epubs.siam.org/doi/book/10.1137/1.9781611971446)</sup> |

## Convergence of real sequences

The central notion is convergence to a limit. A sequence \( (s_n) \) of real numbers converges to a number \( L \) if, by choosing \( n \) sufficiently large, every value \( s_k \) with \( k \ge n \) can be made as close to \( L \) as desired; a sequence diverges if it does not converge<sup>[1](https://mathbooks.unl.edu/Calculus/sec-7-1-sequences.html)</sup>. The formal definition quantifies this precisely: \( \lim_{n \to \infty} s_n = s \) provided that for any \( \varepsilon > 0 \) there is a real number \( N \) such that \( n > N \) implies \( |s_n - s| < \varepsilon \)<sup>[3](https://math.libretexts.org/Bookshelves/Analysis/Real_Analysis_%28Boman_and_Rogers%29/04%3A_Convergence_of_Sequences_and_Series/4.01%3A_Sequences_of_Real_Numbers)</sup>. The condition \( |s_n - s| < \varepsilon \) is equivalent to \( s_n \in (s - \varepsilon, s + \varepsilon) \), so a proof of convergence amounts to answering a challenge: given a distance \( \varepsilon \), produce an index \( N \) beyond which all terms lie within \( \varepsilon \) of the limit<sup>[3](https://math.libretexts.org/Bookshelves/Analysis/Real_Analysis_%28Boman_and_Rogers%29/04%3A_Convergence_of_Sequences_and_Series/4.01%3A_Sequences_of_Real_Numbers)</sup>.

**Worked examples.** To prove \( \lim_{n \to \infty} \frac{n}{n+100} = 1 \), one estimates \( \left| \frac{n}{n+100} - 1 \right| \le \frac{100}{n} \) and chooses \( N = \frac{100}{\varepsilon} \); then \( n > N \) forces the difference below \( \varepsilon \)<sup>[3](https://math.libretexts.org/Bookshelves/Analysis/Real_Analysis_%28Boman_and_Rogers%29/04%3A_Convergence_of_Sequences_and_Series/4.01%3A_Sequences_of_Real_Numbers)</sup>. The same technique shows \( \lim_{n \to \infty} \frac{\sin n}{n} = 0 \), using \( |\sin n| \le 1 \) and \( N = \frac{1}{\varepsilon} \)<sup>[3](https://math.libretexts.org/Bookshelves/Analysis/Real_Analysis_%28Boman_and_Rogers%29/04%3A_Convergence_of_Sequences_and_Series/4.01%3A_Sequences_of_Real_Numbers)</sup>. Not every sequence converges: \( (1 + (-1)^n) = (2, 0, 2, 0, 2, \ldots) \) does not converge to zero, because taking \( \varepsilon = 2 \) defeats every choice of \( N \)<sup>[3](https://math.libretexts.org/Bookshelves/Analysis/Real_Analysis_%28Boman_and_Rogers%29/04%3A_Convergence_of_Sequences_and_Series/4.01%3A_Sequences_of_Real_Numbers)</sup>.

Several standard tools establish convergence without returning to the definition. A bounded monotone sequence converges to a limit<sup>[2](https://ocw.mit.edu/courses/18-100a-real-analysis-fall-2020/pages/lecture-notes-and-readings/)</sup>, and the [squeeze theorem](https://www.edgepedia.org/squeeze-theorem) deduces convergence of a sequence squeezed between two sequences with a common limit<sup>[9](https://math.unm.edu/~crisp/courses/math401/fall17/review2.pdf)</sup>. Limits also interact with order and algebraic operations: the relations between limits and order, algebraic operations, and the absolute value on \( \mathbb{R} \) are standard course material built directly on sequence convergence<sup>[2](https://ocw.mit.edu/courses/18-100a-real-analysis-fall-2020/pages/lecture-notes-and-readings/)</sup>.

## Subsequences, limsup, and liminf

A subsequence \( (a_{n_k}) \) selects terms of a sequence in increasing index order. Subsequence behavior characterizes convergence: \( (a_n) \) converges to \( a \) if and only if each subsequence of \( (a_n) \) converges to \( a \)<sup>[4](https://jeanmariedufour.research.mcgill.ca/ResE/Dufour_1992_C_TS_SequencesSeries.pdf)</sup>.

The [limit superior and limit inferior](https://www.edgepedia.org/limit-superior-and-limit-inferior) capture the extreme cluster behavior of a bounded sequence. For a bounded sequence \( (x_n) \), \( \limsup (x_n) \) is the largest limit approached by subsequences and \( \liminf (x_n) \) the smallest; one can always extract a subsequence converging to the limsup<sup>[10](https://didattica.unibocconi.it/mypage/dwload.php?nomefile=Chapter_220160921124159.pdf)</sup>. These quantities give a clean convergence test: a bounded sequence \( (x_n) \) is convergent if and only if \( \limsup (x_n) = \liminf (x_n) \)<sup>[5](https://faculty.ksu.edu.sa/sites/default/files/0471433314realanalysis41.pdf)</sup>. The same equivalence is stated as: a sequence of real numbers converges if and only if the limsup and the liminf coincide<sup>[9](https://math.unm.edu/~crisp/courses/math401/fall17/review2.pdf)</sup>.

The [Bolzano–Weierstrass theorem](https://www.edgepedia.org/bolzano-weierstrass-theorem) states that every bounded sequence has at least one convergent subsequence<sup>[9](https://math.unm.edu/~crisp/courses/math401/fall17/review2.pdf)</sup><sup> • </sup><sup>[6](https://neckpinch.com/guide/numbers-limits-and-the-integral/sequences/)</sup>. Together with the monotone convergence theorem and the limsup/liminf machinery, it forms the standard toolkit of a first course in real analysis<sup>[2](https://ocw.mit.edu/courses/18-100a-real-analysis-fall-2020/pages/lecture-notes-and-readings/)</sup>.

## Cauchy sequences and completeness

A sequence is Cauchy when its terms eventually lie arbitrarily close to each other, a condition that does not presuppose knowledge of the limit. The convergence theorem for Cauchy sequences states that if a sequence converges, then it converges in the Cauchy sense<sup>[4](https://jeanmariedufour.research.mcgill.ca/ResE/Dufour_1992_C_TS_SequencesSeries.pdf)</sup>. The converse holds in the real numbers and expresses their completeness: a sequence of real numbers converges exactly when it is Cauchy<sup>[6](https://neckpinch.com/guide/numbers-limits-and-the-integral/sequences/)</sup>. This "deep fact" that Cauchy sequences are convergent in \( \mathbb{R} \) is a standard formulation of completeness in analysis courses<sup>[9](https://math.unm.edu/~crisp/courses/math401/fall17/review2.pdf)</sup>.

Completeness also appears in the nested-interval form that Weierstrass preferred: Bolzano's principle that a sequence of nested closed intervals in \( \mathbb{R} \), with \( [a_{m+1}, b_{m+1}] \subset [a_m, b_m] \), "contains" at least one real number, in modern terms has a non-empty intersection<sup>[11](https://plato.stanford.edu/entries/settheory-early/)</sup>.

## Historical development

The modern convergence concept took shape in the early nineteenth century. Bernard Bolzano (1781–1848) essentially formulated the modern criterion of convergence in his *Purely Analytic Proof* of 1817, four years before [Augustin-Louis Cauchy](https://www.edgepedia.org/augustin-louis-cauchy)'s *Cours d'analyse* (1821); that work already contains what is now called the Bolzano–Weierstrass theorem<sup>[7](https://plato.stanford.edu/archives/sum2023/entries/bolzano/)</sup>. Bolzano showed in 1817 that a bounded [Cauchy sequence](https://www.edgepedia.org/cauchy-sequence) of real numbers has a least upper bound, and later worked out an unpublished theory of real numbers defined in terms of convergent sequences of rational numbers<sup>[12](https://mathshistory.st-andrews.ac.uk//HistTopics/Real_numbers_2/)</sup>.

Cauchy (1789–1857) pioneered the study of real and complex analysis and researched convergence and divergence of infinite series, among many other fields<sup>[13](https://mathshistory.st-andrews.ac.uk/Biographies/Cauchy/)</sup>. In the *Cours d'analyse* he stated that a real number is the limit of a sequence of rational numbers, but he treated this as an obvious property rather than a definition, and he did not require the sequence to be Cauchy, which is necessary to define convergence without assuming the limit exists<sup>[12](https://mathshistory.st-andrews.ac.uk//HistTopics/Real_numbers_2/)</sup>.

After Cauchy, the concept of sequence became central to mathematics as late nineteenth-century analysts such as Karl Weierstrass (1815–1897), [Georg Cantor](https://www.edgepedia.org/georg-cantor) (1845–1911), and Julius Wilhelm Dedekind (1831–1916) pursued the concept of the infinite<sup>[14](https://progettomatematica.dm.unibo.it/successioni/introduzionestorica.htm)</sup>. Cantor provided a definition of \( \mathbb{R} \) in 1872 employing Cauchy sequences of rational numbers, an elegant simplification of the definition offered in Weierstrass's lectures<sup>[11](https://plato.stanford.edu/entries/settheory-early/)</sup>. Along a parallel line, Charles Méray considered Cauchy sequences of rational numbers that did not converge to a rational limit and called their limits "fictitious limits", taking the real numbers to consist of the rationals plus these fictitious limits; three years later Heine published a similar, independent notion in his *Elemente der Functionenlehre*, and Heine's system became one of the two standard ways of defining the real numbers today<sup>[12](https://mathshistory.st-andrews.ac.uk//HistTopics/Real_numbers_2/)</sup>.

## Sequences of functions

When the terms of a sequence are functions \( f_1, f_2, f_3, \ldots \) with a common domain, two modes of convergence are distinguished. The sequence converges pointwise to \( f \) if \( f_n(x) \to f(x) \) for every fixed \( x \) in the domain; it converges uniformly if a single stage \( N \), depending only on \( \varepsilon \), works for every \( x \) at once<sup>[15](https://a-warded.org/open-book/university-real-analysis/lesson-31)</sup>. [Uniform convergence](https://www.edgepedia.org/uniform-convergence) always implies pointwise convergence, but the converse is not true<sup>[16](https://www2.math.upenn.edu/~gressman/analysis/06-uniformconv.html)</sup>.

The distinction matters because properties of the terms transfer to the limit only under stronger convergence. If a sequence of continuous functions converges uniformly to \( f \), then \( f \) is continuous<sup>[17](https://connect.gonzaga.edu/asset/file/1558/ch08_123_pointwise_unif_conv_v00.html)</sup>; pointwise limits of continuous functions need not be continuous, and a sequence of polynomials can converge pointwise to a discontinuous function<sup>[18](https://math.csudh.edu/~pong/a3/Notes_08.html)</sup>. The theory is organized around the uniform norm, under which a pointwise convergent sequence is Cauchy in the uniform norm if and only if it converges uniformly<sup>[19](https://ocw.mit.edu/courses/18-100a-real-analysis-fall-2020/mit18_100af20_basic_analysis.pdf)</sup>. Standard results include the Weierstrass criterion, the continuity of uniform limits, and applications to power series<sup>[20](https://ocw.mit.edu/courses/18-100c-real-analysis-fall-2012/resources/mit18_100cf12_l16sum/)</sup>, with the Weierstrass M-test providing a practical test for uniform convergence of series of functions<sup>[2](https://ocw.mit.edu/courses/18-100a-real-analysis-fall-2020/pages/lecture-notes-and-readings/)</sup>.

## Sequences in computation and topology

[Numerical analysis](https://www.edgepedia.org/numerical-analysis) runs on sequences. Iterative methods for linear systems compute a sequence \( x_0, x_1, x_2, \ldots \) of ever better approximate solutions of \( Ax = b \), stopping when \( x_i \) is accurate enough; direct methods such as [Gaussian elimination](https://www.edgepedia.org/gaussian-elimination), by contrast, would give the exact solution after finitely many steps in the absence of roundoff error<sup>[8](https://epubs.siam.org/doi/book/10.1137/1.9781611971446)</sup>. Finite-precision arithmetic adds a qualification: numerical stability or convergence proved in exact arithmetic may become conditional on the arithmetic, because convergence parameters can couple with the computer precision \( \varepsilon \)<sup>[21](https://epubs.siam.org/doi/book/10.1137/1.9780898719673)</sup>.

In general topology, sequences turn out to be insufficient. Convergence of sequences works well when the space is first countable, meaning each point has a countable neighborhood basis, but in spaces without countable bases sequences fall short: they are, in the standard phrase, too short and too thin<sup>[22](https://stijnvermeeren.be/files/mathematics/nets.pdf)</sup>. For example, the discrete topology and the countable complement topology on an uncountable set have exactly the same converging sequences, yet the discrete topology is strictly finer<sup>[22](https://stijnvermeeren.be/files/mathematics/nets.pdf)</sup>. The repair is the net, a generalized sequence indexed by an arbitrary directed set rather than the natural numbers, which is particularly well suited to topological spaces that are not first countable<sup>[23](https://mathworld.wolfram.com/Limit.html)</sup>. In Hausdorff spaces every converging sequence has a unique limit, while a sequence in a space that fails to be Hausdorff may have several, even infinitely many, different limits<sup>[23](https://mathworld.wolfram.com/Limit.html)</sup>.

## References

1. [Sequences, Calculus Guidebook 2A.6, University of Nebraska–Lincoln](https://mathbooks.unl.edu/Calculus/sec-7-1-sequences.html)
2. [Lecture Notes and Readings, 18.100A Real Analysis, MIT OpenCourseWare](https://ocw.mit.edu/courses/18-100a-real-analysis-fall-2020/pages/lecture-notes-and-readings/)
3. [4.1: Sequences of Real Numbers, Real Analysis (Boman and Rogers), Mathematics LibreTexts](https://math.libretexts.org/Bookshelves/Analysis/Real_Analysis_%28Boman_and_Rogers%29/04%3A_Convergence_of_Sequences_and_Series/4.01%3A_Sequences_of_Real_Numbers)
4. [Jean-Marie Dufour (1992). Sequences and Series, McGill University course notes](https://jeanmariedufour.research.mcgill.ca/ResE/Dufour_1992_C_TS_SequencesSeries.pdf)
5. [Real Analysis (excerpt, Theorem 3.4.12), Kansas State University faculty file](https://faculty.ksu.edu.sa/sites/default/files/0471433314realanalysis41.pdf)
6. [Sequences, Neckpinch Guidebook 2A.6](https://neckpinch.com/guide/numbers-limits-and-the-integral/sequences/)
7. [Bernard Bolzano, Stanford Encyclopedia of Philosophy (Summer 2023 edition)](https://plato.stanford.edu/archives/sum2023/entries/bolzano/)
8. [Applied Numerical Linear Algebra, SIAM Publications Library](https://epubs.siam.org/doi/book/10.1137/1.9781611971446)
9. [Review and Practice Problems for Exam 2, MATH 401, University of New Mexico](https://math.unm.edu/~crisp/courses/math401/fall17/review2.pdf)
10. [Sequences: Convergence, Liminf, Limsup and the Cauchy Criterion, Bocconi University course chapter](https://didattica.unibocconi.it/mypage/dwload.php?nomefile=Chapter_220160921124159.pdf)
11. [The Early Development of Set Theory, Stanford Encyclopedia of Philosophy](https://plato.stanford.edu/entries/settheory-early/)
12. [Real numbers 2, MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk//HistTopics/Real_numbers_2/)
13. [Augustin-Louis Cauchy (1789–1857), MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/Biographies/Cauchy/)
14. [Introduzione storica, Progetto Matematica, Università di Bologna](https://progettomatematica.dm.unibo.it/successioni/introduzionestorica.htm)
15. [Pointwise and Uniform Convergence of Function Sequences, A-Warded Real Analysis](https://a-warded.org/open-book/university-real-analysis/lesson-31)
16. [Advanced Analysis: Uniform Convergence, University of Pennsylvania](https://www2.math.upenn.edu/~gressman/analysis/06-uniformconv.html)
17. [Math 413, Sequences of Functions, Gonzaga University](https://connect.gonzaga.edu/asset/file/1558/ch08_123_pointwise_unif_conv_v00.html)
18. [Notes 08: Pointwise and Uniform Convergence, CSU Dominguez Hills](https://math.csudh.edu/~pong/a3/Notes_08.html)
19. [Basic Analysis: Introduction to Real Analysis, MIT 18.100A course text](https://ocw.mit.edu/courses/18-100a-real-analysis-fall-2020/mit18_100af20_basic_analysis.pdf)
20. [18.100C Real Analysis: Lecture 16 Summary, MIT OpenCourseWare](https://ocw.mit.edu/courses/18-100c-real-analysis-fall-2012/resources/mit18_100cf12_l16sum/)
21. [Lectures on Finite Precision Computations, SIAM Publications Library](https://epubs.siam.org/doi/book/10.1137/1.9780898719673)
22. [Sequences and nets in topology, Stijn Vermeeren](https://stijnvermeeren.be/files/mathematics/nets.pdf)
23. [Limit, Wolfram MathWorld](https://mathworld.wolfram.com/Limit.html)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Analysis and mathematical models › Real analysis*

*Initially written Sep 17, 2026 · Reviewed: Sep 30, 2026; Oct 15, 2026 · Edited: Sep 30, 2026; Oct 15, 2026 · Last review: Oct 15, 2026*

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