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]