A **sequence** is a function from the natural numbers into a [set](/page/Set) — an ordered list $a_1, a_2, a_3, \ldots$ indexed by $\mathbb{N}$. Sequences are the simplest infinite objects in mathematics, and they are the vehicle through which the concept of [limit](/page/Limit) enters analysis. Every major construction in analysis — derivatives, [integrals](/page/Integral), [series](/page/Series), solutions to differential equations — is defined as the limit of a sequence (of difference quotients, Riemann sums, partial sums, or iterates), and the properties of these constructions depend on the convergence behaviour of the underlying sequences. This page develops sequences as mathematical objects: the formal definition, the taxonomy of convergence behaviour (convergent, divergent, bounded, monotone, Cauchy), the algebra and order structure of convergent sequences, subsequences and limit points, the role of sequences in characterising topological properties ([closed sets](/page/Closed%20Set), [continuity](/page/Continuous), compactness), and sequences of [functions](/page/Function). [motivation] ## Motivation ### Why Sequences? The natural numbers $1, 2, 3, \ldots$ are the simplest infinite set, and a sequence is the simplest way to organise infinitely many objects: line them up and index them by $\mathbb{N}$. This indexing provides an *order* — there is a first term, a second term, a hundredth term — and this order is what makes convergence possible. A set $\{1/n : n \in \mathbb{N}\}$ contains points near $0$, but it is the *sequence* $1, 1/2, 1/3, \ldots$ that converges to $0$, because the ordering tells us that the terms get closer to $0$ as the index grows. ### Sequences as the Bridge to the Infinite Analysis is the mathematics of the infinite — infinite sums, infinite products, limits of infinite processes. But every concrete computation involves only finitely many steps. Sequences bridge this gap: an infinite process is encoded as a sequence of finite approximations, and the [limit](/page/Limit) of the sequence (if it exists) is the result of the infinite process. The decimal expansion $3.14159\ldots$ is the sequence of finite decimals $3, 3.1, 3.14, 3.141, \ldots$ converging to $\pi$. The exponential function $e^x$ is the limit of the sequence of partial sums $\sum_{k=0}^n x^k/k!$. The solution to an ODE is the limit of the Picard iterates. ### Sequences vs. Nets and Filters In [metric spaces](/page/Metric%20Space), sequences suffice to characterise all topological properties: a set is [closed](/page/Closed%20Set) if and only if it is closed under sequential limits, a function is [continuous](/page/Continuous) if and only if it preserves sequential limits, and compactness is equivalent to sequential compactness. In general [topological spaces](/page/Topology), sequences are insufficient — the correct generalisations are *nets* (sequences indexed by directed sets) and *filters*. This page focuses on sequences; the generalisations are discussed on the [Topology](/page/Topology) page. [/motivation] ## Definition A sequence is a function from $\mathbb{N}$ into a set. The set can be $\mathbb{R}$, $\mathbb{C}$, a metric space, a [topological space](/page/Topology), or any mathematical structure. [definition:Sequence] Let $X$ be a set. A **sequence** in $X$ is a function $a: \mathbb{N} \to X$. The value $a(n)$ is called the **$n$-th term** and is written $a_n$. The sequence itself is denoted $\{a_n\}_{n=1}^\infty$, $(a_n)_{n=1}^\infty$, or simply $\{a_n\}$. [/definition] Despite the notation $\{a_n\}$ (which uses set braces), a sequence is *not* a set — the ordering matters, and repetitions are allowed. The sequence $1, 1, 1, \ldots$ (the constant sequence) and the sequence $1, 2, 1, 2, \ldots$ are different sequences, even though their ranges as sets ($\{1\}$ and $\{1, 2\}$) are "simpler" objects. The **range** (or **image**) of a sequence $\{a_n\}$ is the set $\{a_n : n \in \mathbb{N}\} \subseteq X$, which may be finite even if the sequence is infinite (as the constant sequence shows). [example:Sequences In Various Spaces] The sequence $a_n = 1/n$ is a sequence in $\mathbb{R}$. The sequence $z_n = e^{2\pi i/n}$ is a sequence in $\mathbb{C}$ (and on the unit circle $S^1$). The sequence $f_n(x) = x^n$ is a sequence in $C([0,1])$ (the space of continuous functions on $[0,1]$). The sequence $e_n = (0, \ldots, 0, 1, 0, \ldots)$ (with $1$ in the $n$-th position) is a sequence in $\ell^2$. [/example] ## Convergence and Divergence The central question about any sequence is whether it converges — whether the terms approach a definite [limit](/page/Limit) as the index grows. The precise definition, the $\varepsilon$-$N$ formulation, is developed on the [Limit](/page/Limit) page; here we focus on the taxonomy and behaviour of convergent and divergent sequences. ### Convergent Sequences A sequence $\{a_n\}$ in a metric space $(X, d)$ converges to $L \in X$ if $d(a_n, L) \to 0$ — for every $\varepsilon > 0$, eventually $d(a_n, L) < \varepsilon$. In $\mathbb{R}$, this means $|a_n - L| \to 0$. The limit is [unique](/theorems/625) in any Hausdorff space. Every convergent sequence is bounded: if $a_n \to L$, then $d(a_n, L) < 1$ for $n \ge N$, so $d(a_n, L) \le \max(d(a_1, L), \ldots, d(a_{N-1}, L), 1)$ for all $n$. The converse is false: $a_n = (-1)^n$ is bounded but divergent. The algebra of limits makes the set of convergent real sequences into a vector space: if $a_n \to L$ and $b_n \to M$, then $\alpha a_n + \beta b_n \to \alpha L + \beta M$ for any $\alpha, \beta \in \mathbb{R}$. The limit functional $\lim: c \to \mathbb{R}$ (where $c$ denotes the space of convergent sequences) is a bounded linear functional with $\|\lim\| = 1$. ### Divergent Sequences and Their Types A sequence that does not converge is divergent. Divergence takes several forms: **Divergence to infinity:** $a_n \to +\infty$ means for every $M > 0$, eventually $a_n > M$. Similarly $a_n \to -\infty$. The sequence $a_n = n^2$ diverges to $+\infty$; the sequence $a_n = -\sqrt{n}$ diverges to $-\infty$. **Oscillation:** the sequence $a_n = (-1)^n$ is bounded but has no limit — it oscillates between $-1$ and $1$ forever. More subtly, $a_n = \sin(n)$ is bounded and has dense range in $[-1, 1]$, but no limit. **Unbounded oscillation:** $a_n = n \cdot (-1)^n$ is unbounded and oscillates — it does not diverge to $+\infty$ or $-\infty$ (the subsequence of even terms goes to $+\infty$, the odd terms to $-\infty$). [example:The Sequence Sin N] The sequence $a_n = \sin(n)$ (where $n$ is in radians) is bounded ($|a_n| \le 1$) but divergent. The divergence follows from the fact that $\pi$ is irrational: the fractional parts of $n/(2\pi)$ are equidistributed modulo $1$ (by Weyl's equidistribution theorem), so the sequence $\sin(n)$ is dense in $[-1, 1]$ and has every point of $[-1, 1]$ as a subsequential limit. [/example] ## Bounded and Monotone Sequences The most tractable sequences in $\mathbb{R}$ are those that are bounded, monotone, or both. Monotonicity reduces the convergence question to a boundedness check (the [Monotone Convergence Theorem](/theorems/626)), and boundedness guarantees the existence of convergent subsequences (the [Bolzano-Weierstrass theorem](/theorems/628)). ### Bounded Sequences A sequence $\{a_n\}$ in $\mathbb{R}$ is **bounded above** if $\sup_n a_n < \infty$, **bounded below** if $\inf_n a_n > -\infty$, and **bounded** if both. Equivalently, $\{a_n\}$ is bounded if there exists $M > 0$ with $|a_n| \le M$ for all $n$. Two fundamental quantities associated with a bounded sequence are its limit superior and limit inferior. [definition:Limit Superior And Limit Inferior] Let $\{a_n\}$ be a bounded sequence of real numbers. The **limit superior** (or **lim sup**) is \begin{align*} \limsup_{n \to \infty} a_n := \lim_{n \to \infty} \sup_{k \ge n} a_k = \inf_{n \ge 1} \sup_{k \ge n} a_k. \end{align*} The **limit inferior** (or **lim inf**) is \begin{align*} \liminf_{n \to \infty} a_n := \lim_{n \to \infty} \inf_{k \ge n} a_k = \sup_{n \ge 1} \inf_{k \ge n} a_k. \end{align*} [/definition] The lim sup is the largest subsequential limit, and the lim inf is the smallest. A bounded sequence converges if and only if $\limsup a_n = \liminf a_n$, and the common value is the limit. The lim sup and lim inf always exist for bounded sequences (they are limits of monotone bounded sequences — $\sup_{k \ge n} a_k$ is decreasing and bounded below, $\inf_{k \ge n} a_k$ is increasing and bounded above — so the [Monotone Convergence Theorem](/theorems/626) applies). ### Monotone Sequences and the Completeness of $\mathbb{R}$ A sequence is **increasing** if $a_n \le a_{n+1}$ for all $n$, **decreasing** if $a_n \ge a_{n+1}$, and **monotone** if it is either increasing or decreasing. The Monotone Convergence Theorem says that bounded monotone sequences converge: [quotetheorem:626] This result is equivalent to the completeness of $\mathbb{R}$ (the least upper bound property). It fails in $\mathbb{Q}$: the increasing sequence of decimal approximations to $\sqrt{2}$ is bounded above by $2$ but has no limit in $\mathbb{Q}$. The theorem is the primary tool for proving convergence when the limit is unknown — one establishes monotonicity and boundedness, and the existence of the limit follows without needing to identify it. [example:Recursive Sequence Via Monotone Convergence] Define $a_1 = 2$ and $a_{n+1} = \frac{1}{2}(a_n + 6/a_n)$. The AM-GM inequality gives $a_{n+1} \ge \sqrt{6}$ for $n \ge 1$. Once $a_n \ge \sqrt{6}$, we have $a_{n+1} - a_n = (6 - a_n^2)/(2a_n) \le 0$, so the sequence is eventually decreasing. By the [Monotone Convergence Theorem](/theorems/626), $\{a_n\}$ converges to some $L \ge \sqrt{6}$. Taking limits in the recursion: $L = (L + 6/L)/2$, so $L^2 = 6$, giving $L = \sqrt{6}$. [/example] ## Subsequences and Limit Points A subsequence extracts infinitely many terms from a sequence while preserving the original order. Subsequences detect what a sequence "tries to do" even when it fails to converge — the set of all subsequential limits describes the long-term behaviour of the sequence completely. ### Subsequences [definition:Subsequence] Let $\{a_n\}_{n=1}^\infty$ be a sequence. A **subsequence** of $\{a_n\}$ is a sequence $\{a_{n_k}\}_{k=1}^\infty$ where $n_1 < n_2 < n_3 < \cdots$ is a strictly increasing sequence of natural numbers. [/definition] The condition $n_1 < n_2 < \cdots$ ensures that a subsequence passes through the terms of the original sequence in order, skipping some but never rearranging. This forces $n_k \ge k$ for all $k$, so a subsequence "goes to infinity" at least as fast as the original. If $a_n \to L$, then every subsequence $a_{n_k} \to L$ (the tail condition $|a_n - L| < \varepsilon$ for $n \ge N$ applies to $n_k$ since $n_k \ge k \ge N$). The converse is useful for proving convergence: if every subsequence of $\{a_n\}$ has a further subsequence converging to $L$, then $a_n \to L$. ### The Bolzano-Weierstrass Theorem The fundamental result about subsequences is that bounded sequences always have convergent subsequences: [quotetheorem:628] This is the sequential formulation of compactness for $\mathbb{R}^n$: a subset $K \subseteq \mathbb{R}^n$ is [compact](/page/Topology) if and only if every sequence in $K$ has a subsequence converging to a point of $K$. The proof by bisection is constructive and provides an explicit method for extracting convergent subsequences. ### Limit Points of a Sequence A **limit point** (or **cluster point**, or **accumulation point**) of a sequence $\{a_n\}$ is a value $L$ such that some subsequence converges to $L$. Equivalently, $L$ is a limit point if for every $\varepsilon > 0$ and every $N \in \mathbb{N}$, there exists $n \ge N$ with $|a_n - L| < \varepsilon$ — the sequence returns to every neighbourhood of $L$ infinitely often. The set of all limit points of a bounded sequence is nonempty (by Bolzano-Weierstrass), [closed](/page/Closed%20Set), and bounded. The $\limsup$ and $\liminf$ are respectively the largest and smallest elements of this set. A sequence converges if and only if it has exactly one limit point (and is bounded — the sequence $1, 1, 2, 1, 3, 1, 4, 1, \ldots$ has $1$ as its only limit point but diverges because the terms $2, 3, 4, \ldots$ are unbounded). [example:Limit Points Of An Oscillating Sequence] The sequence $a_n = \sin(n\pi/4)$ takes the values $\frac{\sqrt{2}}{2}, 1, \frac{\sqrt{2}}{2}, 0, -\frac{\sqrt{2}}{2}, -1, -\frac{\sqrt{2}}{2}, 0, \frac{\sqrt{2}}{2}, 1, \ldots$ (period $8$). The set of limit points is $\{0, \pm\frac{\sqrt{2}}{2}, \pm 1\}$ — exactly the range of the sequence, since every value is revisited infinitely often. The $\limsup$ is $1$ and the $\liminf$ is $-1$. [/example] ## [Cauchy Sequences](/page/Cauchy%20Sequence) and Completeness The convergence definition requires knowing the limit $L$. The Cauchy criterion provides an *intrinsic* test: it detects convergence by asking whether the terms get close to *each other*, without reference to any external limit. ### The Cauchy Criterion [definition:Cauchy Sequence] A sequence $\{a_n\}$ in a metric space $(X, d)$ is **Cauchy** if for every $\varepsilon > 0$ there exists $N \in \mathbb{N}$ with \begin{align*} m, n \ge N \implies d(a_m, a_n) < \varepsilon. \end{align*} [/definition] Every convergent sequence is Cauchy (triangle inequality). The converse — every Cauchy sequence converges — is the definition of a **complete** metric space. In $\mathbb{R}$ (and $\mathbb{R}^n$), completeness holds: [quotetheorem:172] This result, often called the **General Principle of Convergence**, is equivalent to the least upper bound property of $\mathbb{R}$ and to the [Monotone Convergence Theorem](/theorems/626) and the [Bolzano-Weierstrass theorem](/theorems/628). Together, these four equivalent formulations of completeness form the foundation on which all of real analysis is built. The Cauchy criterion is indispensable when the limit is not known explicitly. To show that $\sum_{k=1}^\infty 1/k^2$ converges, one does not need to know the sum is $\pi^2/6$; it suffices to show that the partial sums $S_n = \sum_{k=1}^n 1/k^2$ are Cauchy, which follows from the estimate $|S_m - S_n| \le \sum_{k=n+1}^m 1/k^2 \le \int_n^m x^{-2} \, d\mathcal{L}^1 = 1/n - 1/m < 1/n \to 0$. ### Completeness and Incompleteness Completeness depends on the ambient space, not on the sequence. The sequence of decimal approximations to $\sqrt{2}$ is Cauchy in both $\mathbb{Q}$ and $\mathbb{R}$, but converges only in $\mathbb{R}$. The space $\mathbb{Q}$ is not complete — it has "gaps" where limits of Cauchy sequences should be. The construction of $\mathbb{R}$ from $\mathbb{Q}$ (via Dedekind cuts or equivalence classes of Cauchy sequences) is precisely the process of filling these gaps. In analysis, the most important complete spaces are the [Banach spaces](/page/Banach%20Space) — complete [normed vector spaces](/page/Normed%20Vector%20Space). The completeness of $L^p$ spaces, [Sobolev spaces](/page/Sobolev%20Space), and the space $C(K)$ with the supremum norm is what makes existence theorems (Banach fixed point, Lax-Milgram, spectral decomposition) possible. ## Sequences and Topology In metric spaces, sequences are the primary tool for detecting topological properties. Every concept in point-set topology — open, closed, closure, continuity, compactness — has a sequential characterisation that is often the most efficient way to work with it. ### Sequential Characterisation of Closed Sets A subset $F$ of a metric space is [closed](/page/Closed%20Set) if and only if it contains the limits of all its convergent sequences: $\{a_n\} \subseteq F$ and $a_n \to L$ implies $L \in F$. This is the working definition in most analytic arguments — to show $F$ is closed, take an arbitrary convergent sequence in $F$ and show the limit is in $F$. The closure $\overline{A}$ of a set $A$ in a metric space consists precisely of the limits of sequences in $A$: $x \in \overline{A}$ if and only if there exists a sequence $\{a_n\} \subseteq A$ with $a_n \to x$. ### [Sequential Characterisation of Continuity](/theorems/285) A function $f: X \to Y$ between metric spaces is [continuous](/page/Continuous) at $a$ if and only if $a_n \to a$ implies $f(a_n) \to f(a)$. This is the sequential definition of continuity, and it is equivalent to the $\varepsilon$-$\delta$ definition and the topological (open-preimage) definition. ### Sequential Compactness A metric space is [compact](/page/Topology) if and only if every sequence has a convergent subsequence (sequential compactness). In $\mathbb{R}^n$, this is the content of the [Bolzano-Weierstrass theorem](/theorems/628) combined with the [Heine-Borel theorem](/theorems/315): a subset of $\mathbb{R}^n$ is compact if and only if it is closed and bounded, if and only if every sequence in it has a convergent subsequence with limit in the set. [example:Sequential Argument For Closedness] The set $F = \{(x, y) \in \mathbb{R}^2 : xy \ge 1,\; x > 0\}$ is closed. Let $(x_n, y_n) \in F$ with $(x_n, y_n) \to (a, b)$. Then $x_n > 0$ and $x_n y_n \ge 1$ for all $n$. Since $x_n \to a$, we have $a \ge 0$. If $a = 0$, then $y_n \ge 1/x_n \to \infty$, contradicting $y_n \to b < \infty$. So $a > 0$, and $ab = \lim x_n y_n \ge 1$ (limits preserve non-strict inequalities). Therefore $(a, b) \in F$. [/example] ## Sequences of Functions When the terms of a sequence are functions rather than numbers, new phenomena arise — the interplay between convergence in the index $n$ and the behaviour in the variable $x$ gives rise to different notions of convergence (pointwise, uniform, $L^p$) with different properties. ### Pointwise and Uniform Convergence A sequence of functions $f_n: E \to \mathbb{R}$ converges **pointwise** to $f$ if $f_n(x) \to f(x)$ for each $x \in E$. It converges **uniformly** to $f$ if $\sup_{x \in E} |f_n(x) - f(x)| \to 0$. [Uniform convergence](/page/Uniform%20Convergence) implies pointwise convergence; the converse is false. [definition:Uniform Convergence] A sequence of functions $f_n: E \to \mathbb{R}$ converges **uniformly** to $f: E \to \mathbb{R}$ if for every $\varepsilon > 0$ there exists $N \in \mathbb{N}$ such that \begin{align*} n \ge N \implies |f_n(x) - f(x)| < \varepsilon \quad \text{for all } x \in E. \end{align*} Equivalently, $\|f_n - f\|_\infty := \sup_{x \in E} |f_n(x) - f(x)| \to 0$. [/definition] The key distinction: in pointwise convergence, the index $N$ may depend on both $\varepsilon$ and the point $x$; in uniform convergence, $N$ depends only on $\varepsilon$. This "uniformity" is what makes the limit function well-behaved. ### What Uniform Convergence Preserves The uniform limit of [continuous](/page/Continuous) functions is continuous (the "$\varepsilon/3$ argument": $|f(x) - f(y)| \le |f(x) - f_n(x)| + |f_n(x) - f_n(y)| + |f_n(y) - f(y)|$, and each term can be made small). Pointwise limits of continuous functions need not be continuous: the sequence $f_n(x) = x^n$ on $[0, 1]$ converges pointwise to $\mathbb{1}_{\{1\}}$, which is discontinuous. Uniform convergence also commutes with integration on bounded domains ($\lim \int f_n = \int \lim f_n$) and, under additional hypotheses, with [differentiation](/page/Derivative). The [Cauchy Criterion for Uniform Convergence](/theorems/257) provides a way to verify uniform convergence without knowing the limit. [example:Pointwise But Not Uniform Convergence] The sequence $f_n(x) = x^n$ on $[0, 1]$ converges pointwise to $f(x) = 0$ for $0 \le x < 1$ and $f(1) = 1$. The convergence is not uniform: $\sup_{x \in [0,1]} |f_n(x) - f(x)| \ge |f_n(1 - 1/n) - 0| = (1 - 1/n)^n \to 1/e \neq 0$. The limit function is discontinuous at $x = 1$, which is impossible for a uniform limit of continuous functions — confirming that convergence cannot be uniform. [/example] [example:Uniform Convergence And Continuity] Define $g_n(x) = \frac{x}{1 + nx^2}$ on $\mathbb{R}$. For each fixed $x \neq 0$, $|g_n(x)| \le 1/(n|x|) \to 0$, and $g_n(0) = 0$, so $g_n \to 0$ pointwise. The convergence is also uniform: by AM-GM, $1 + nx^2 \ge 2\sqrt{n}|x|$, so $|g_n(x)| \le 1/(2\sqrt{n}) \to 0$. Since each $g_n$ is continuous and the convergence is uniform, the limit $g \equiv 0$ is continuous — as guaranteed by the [uniform limit theorem](/theorems/258). [/example] ## References - Rudin, W., *Principles of Mathematical Analysis* (3rd ed., 1976). - Abbott, S., *Understanding Analysis* (2nd ed., 2015). - Bartle, R. G. and Sherbert, D. R., *Introduction to Real Analysis* (4th ed., 2011). - Munkres, J. R., *Topology* (2nd ed., 2000).