# Limit of a sequence
The limit of a sequence is the number L that the terms of an infinite sequence approach and stay near: a sequence \(\{a_n\}\) converges to L if, for every tolerance \(\epsilon > 0\), there is an integer N such that \(|a_n - L| < \epsilon\) for all \(n \ge N\).[1](https://openstax.org/books/calculus-volume-2/pages/5-1-sequences) When no such number exists, the sequence diverges and the limit does not exist.[1](https://openstax.org/books/calculus-volume-2/pages/5-1-sequences)
| Key fact | Detail |
|---|---|
| Definition | \(\lim_{n \to \infty} a_n = L\) means: for all \(\epsilon > 0\), there exists an integer N with \(|a_n - L| < \epsilon\) whenever \(n \ge N\)[1](https://openstax.org/books/calculus-volume-2/pages/5-1-sequences) |
| Tail behavior | Convergence depends only on what happens as \(n \to \infty\); changing finitely many terms preserves both convergence and the limit[1](https://openstax.org/books/calculus-volume-2/pages/5-1-sequences) |
| Uniqueness | In metric spaces limits are unique in commonly encountered settings; in a non-Hausdorff topological space a sequence can have several, even infinitely many, limits[2](https://mathworld.wolfram.com/Limit.html) |
| Cauchy criterion | A sequence converges if and only if it is Cauchy, meaning \(|s_n - s_m| < \epsilon\) for all \(m, n > N\); this holds in \(\mathbb{R}\) but not in \(\mathbb{Q}\)[3](https://www.math.purdue.edu/~gcavigli/Swanson.pdf) • [4](https://notes.amittai.studio/real-analysis/sequences-series/cauchy-completeness) |
| limsup/liminf test | A sequence converges to \(\ell\) if and only if \(\limsup a_n = \liminf a_n = \ell\)[5](https://www2.math.upenn.edu/~ancoop/3600/section-10.html) |
| Divergence test | If a sequence has subsequences converging to different limits, the sequence diverges[6](https://www.math.ucdavis.edu/~hunter/intro_analysis_pdf/intro_analysis.pdf) |
## The formal definition
The definition quantifies over two variables. The tolerance \(\epsilon\) is chosen first, by an adversary who wants to challenge the claim of convergence; the index N is then produced in response, and it may depend on \(\epsilon\). The sequence converges to \(L\) if, for every positive \(\epsilon\), there exists an integer N (which may depend on \(\epsilon\)) such that past index N all terms lie within the band \((L - \epsilon, L + \epsilon)\).[1](https://openstax.org/books/calculus-volume-2/pages/5-1-sequences) • [15](https://math.libretexts.org/Courses/De_Anza_College/Calculus_III%3A_Series_and_Vector_Calculus/01%3A_Sequences_and_Series/1.01%3A_Sequences) Because the condition constrains only the tail, inserting finitely many arbitrary terms \(b_1, \dots, b_N\) before \(a_1\) changes nothing: the new sequence converges if and only if the original does, and to the same limit.[1](https://openstax.org/books/calculus-volume-2/pages/5-1-sequences)
Two standard examples show the definition at work. The sequence \(\{1/n\}\) converges to 0, which can be read off from the function \(f(x) = 1/x\), since if \(\lim_{x \to \infty} f(x)\) exists for a function with \(f(n) = a_n\), the sequence converges to the same limit.[1](https://openstax.org/books/calculus-volume-2/pages/5-1-sequences) The sequence \(\{(-1)^n\}\) diverges because its terms alternate between 1 and −1 and never settle near a single value.[1](https://openstax.org/books/calculus-volume-2/pages/5-1-sequences)
In a general topological space the definition takes a neighborhood form: a sequence \(\{x_n\}\) in a space X has limit x if, for every neighborhood U of x, there is an N such that \(x_n \in U\) for all \(n \ge N\).[2](https://mathworld.wolfram.com/Limit.html) Specialized to a metric space this becomes \(|x_n - L| < \epsilon\) for all \(n \ge n_0\).[2](https://mathworld.wolfram.com/Limit.html)
## The Cauchy criterion and completeness
The \(\epsilon\)-N definition names the limit L, which is a problem when the limit is not yet available to name: a decimal expansion, a Newton iteration, or the partial sums of a series are all described by sequences approaching a number one wants to prove exists first. The Cauchy criterion removes L from the statement. A sequence is Cauchy if for every \(\epsilon > 0\) there is an N such that \(|s_n - s_m| < \epsilon\) for all \(m, n > N\), so the terms eventually bunch together within any tolerance of one another.[3](https://www.math.purdue.edu/~gcavigli/Swanson.pdf) • [4](https://notes.amittai.studio/real-analysis/sequences-series/cauchy-completeness)
**Convergence implies Cauchy in any ordered field.** If \(x_n \to x\), then for \(n, k \ge M\) the triangle inequality gives \(|x_n - x_k| \le |x_n - x| + |x - x_k| < \epsilon/2 + \epsilon/2 = \epsilon\).[4](https://notes.amittai.studio/real-analysis/sequences-series/cauchy-completeness) The converse direction is the completeness of the real numbers: a sequence of real numbers is Cauchy if and only if it converges.[4](https://notes.amittai.studio/real-analysis/sequences-series/cauchy-completeness) One proof route shows a [Cauchy sequence](https://www.edgepedia.org/cauchy-sequence) is bounded, so its [limit superior and limit inferior](https://www.edgepedia.org/limit-superior-and-limit-inferior) exist by the least-upper-bound property, and then shows these two quantities coincide.[4](https://notes.amittai.studio/real-analysis/sequences-series/cauchy-completeness) The chain of equivalences runs from the least-upper-bound property through the Bolzano-Weierstrass theorem to the convergence of every Cauchy sequence.[7](https://math.csudh.edu/~pong/a1/Week_08.html)
The distinction between \(\mathbb{R}\) and \(\mathbb{Q}\) is exactly here. The metric space \(\mathbb{Q}\) is not complete, meaning it contains Cauchy sequences that do not converge to a rational limit, while \(\mathbb{R}\) and \(\mathbb{R}^d\) are complete: every Cauchy sequence in them converges.[8](https://www.math.cmu.edu/~gautam/c/2026-269/notes/sequences.html) A metric space is called complete when every Cauchy sequence in it converges, and any closed subset of a complete metric space is complete in its own right.[8](https://www.math.cmu.edu/~gautam/c/2026-269/notes/sequences.html) This is why the theorem is a limit-free test: to prove a sequence converges, show its terms bunch, and existence of the limit is guaranteed before the limit is identified.[4](https://notes.amittai.studio/real-analysis/sequences-series/cauchy-completeness)
## Subsequences, limsup, and liminf
Every subsequence of a convergent sequence converges to the limit of the sequence, and the limit set, the set of limits of all convergent subsequences, consists of a single point for a convergent sequence.[6](https://www.math.ucdavis.edu/~hunter/intro_analysis_pdf/intro_analysis.pdf) The contrapositive gives a practical divergence test: the sequence \(((-1)^{n+1})\), whose limit set is \(\{-1, 1\}\), diverges because it has subsequences converging to different limits.[6](https://www.math.ucdavis.edu/~hunter/intro_analysis_pdf/intro_analysis.pdf)
The limit superior and limit inferior organize this behavior for any bounded sequence. They are defined as \(\limsup_{n \to \infty} a_n = \inf_n \sup_{k \ge n} a_k\) and \(\liminf_{n \to \infty} a_n = \sup_n \inf_{k \ge n} a_k\), and for a bounded sequence \(\liminf a_n \le \limsup a_n\).[8](https://www.math.cmu.edu/~gautam/c/2026-269/notes/sequences.html) The two are equal if and only if the sequence converges, in which case all three quantities coincide.[8](https://www.math.cmu.edu/~gautam/c/2026-269/notes/sequences.html) Equivalently, \(\limsup a_n\) is the largest and \(\liminf a_n\) the smallest limit of any convergent subsequence, and subsequences realizing each exist.[9](https://math.csudh.edu/~pong/a1/Week_07.html)
## Historical development
The modern definition took shape over the nineteenth century. Bernard Bolzano (1781–1848) was the first to give a rigorous \(\epsilon\)-\(\delta\) definition of a limit, in work that was not widely disseminated at the time; [Augustin-Louis Cauchy](https://www.edgepedia.org/augustin-louis-cauchy) made mathematicians think more deeply about rigor, and Karl Weierstrass (1815–1897), generally regarded as the father of modern analysis, gave the first rigorous \(\epsilon\)-\(\delta\) definition of continuity.[10](https://www.amsi.org.au/backup_ESA_2018.01.12/ESA_Senior_Years/SeniorTopic3/3a/3a_4history_4.html) The symbols \(\epsilon\) and \(\delta\) were introduced by Cauchy in 1823, but Cauchy never specified a functional relationship between them; the epsilon-delta method appeared in full in Weierstrass's definition of a limit in his 1861 lectures.[11](https://arxiv.org/abs/1502.06942) Cauchy's own convergence criterion, published in his 1821 *Cours d'analyse*, rested on geometrical intuition about the continuity of a straight line rather than on a completed theory of the real numbers.[12](https://mathvoices.ams.org/featurecolumn/2025/06/01/irrationality-tamed/)
## Applications: iterative methods
[Numerical analysis](https://www.edgepedia.org/numerical-analysis) supplies the most common practical setting. In fixed-point iteration one rewrites an equation as \(x = g(x)\) and generates \(x_{n+1} = g(x_n)\) from a starting value \(x_0\). If g is continuously differentiable on \([a, b]\), maps \([a, b]\) into itself, and \(k = \max_{a \le x \le b} |g'(x)| < 1\), the iteration converges to a unique fixed point \(\alpha\) for any starting approximation, with the error estimate
\[ |\alpha - x_n| \le \frac{k^n}{1 - k}\,|x_1 - x_0|. \]
The smaller k is, the faster the convergence; when k is close to 1 convergence can be very slow.[13](https://faculty.ksu.edu.sa/sites/default/files/Ch2_lecture2_1st_1446H_0.pdf) Convergence proofs for such iterations proceed by showing the generated sequence is Cauchy, which in the complete space \(\mathbb{R}\) guarantees a limiting point \(x^*\).[14](https://www.cs.cornell.edu/courses/cs4220/2026sp/lec/2026-03-27.html) When \(g\) is continuously differentiable, a Taylor expansion around the fixed point shows the asymptotic error satisfies \(e^{k+1} = g'(x^*)\,e^k + O(\|e^k\|^2)\), and when \(g'(x^*) = 0\) the error at each step behaves like the square of the previous error, a condition called quadratic convergence.[14](https://www.cs.cornell.edu/courses/cs4220/2026sp/lec/2026-03-27.html)
## References
1. [5.1 Sequences, Calculus Volume 2, OpenStax](https://openstax.org/books/calculus-volume-2/pages/5-1-sequences)
2. [Limit, Wolfram MathWorld](https://mathworld.wolfram.com/Limit.html)
3. [Section 8.7: Cauchy sequences, completeness of R, Purdue University](https://www.math.purdue.edu/~gcavigli/Swanson.pdf)
4. [Cauchy Sequences and the Completeness of the Reals, Amittai study notes](https://notes.amittai.studio/real-analysis/sequences-series/cauchy-completeness)
5. [Limits superior and inferior, University of Pennsylvania](https://www2.math.upenn.edu/~ancoop/3600/section-10.html)
6. [Introduction to Analysis, 3. Sequences, UC Davis](https://www.math.ucdavis.edu/~hunter/intro_analysis_pdf/intro_analysis.pdf)
7. [Week 8 notes, CSU Dominguez Hills](https://math.csudh.edu/~pong/a1/Week_08.html)
8. [Sequences and Series lecture notes, Carnegie Mellon University](https://www.math.cmu.edu/~gautam/c/2026-269/notes/sequences.html)
9. [Week 7 notes, CSU Dominguez Hills](https://math.csudh.edu/~pong/a1/Week_07.html)
10. [History and applications: Cauchy and Weierstrass, AMSI](https://www.amsi.org.au/backup_ESA_2018.01.12/ESA_Senior_Years/SeniorTopic3/3a/3a_4history_4.html)
11. [On the history of epsilontics, arXiv](https://arxiv.org/abs/1502.06942)
12. [Irrationality Tamed, AMS Feature Column](https://mathvoices.ams.org/featurecolumn/2025/06/01/irrationality-tamed/)
13. [Numerical Methods, Chapter 2, King Saud University](https://faculty.ksu.edu.sa/sites/default/files/Ch2_lecture2_1st_1446H_0.pdf)
14. [CS 4220 Numerical Analysis, Cornell University](https://www.cs.cornell.edu/courses/cs4220/2026sp/lec/2026-03-27.html)
15. [1.1: Sequences](https://math.libretexts.org/Courses/De_Anza_College/Calculus_III%3A_Series_and_Vector_Calculus/01%3A_Sequences_and_Series/1.01%3A_Sequences)
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