Limit of a sequence

n	n sin(1/n)
1	0.841471
2	0.958851
...
10	0.998334
...
100	0.999983

As the positive integer n becomes larger and larger, the value n sin(1/n) becomes arbitrarily close to 1. We say that "the limit of the sequence n sin(1/n) equals 1."

In mathematics, a limit of a sequence is a value that the terms of the sequence "get close to eventually". If such a limit exists, the sequence converges.

Limits can be defined in any metric or topological space, but are usually first encountered in the real numbers.

Convergence of sequences is a fundamental notion in mathematical analysis, which has been studied since ancient times.

Real numbers

Definition

A real number x is the limit of the sequence (x_n) if the following condition holds:

for each ε > 0, there exists a natural number N such that, for every $n \geq N$, we have $|x_n - x| < \epsilon$.

In other words, for every measure of closeness ε, the sequence's terms are eventually that close to the limit. The sequence (x_n) is said to converge to or tend to the limit x, written $x_n \to x$ or $\lim_{n \to \infty} x_n = x$.

If a sequence converges to some limit, then it is convergent; otherwise it is divergent.

Examples

If $x_n = c$ for some constant c, then $x_n \to c$.

If $x_n = 1/n$, then $x_n \to 0$.

If $x_n = 1/n$ when $n$ is even, and $x_n = 1/n^2$ when $n$ is odd, then $x_n \to 0$. (The fact that $x_{n+1} > x_n$ whenever $n$ is odd is irrelevant.)

Given any real number, one may easily construct a sequence that converges to that number by taking decimal approximations. For example, the sequence $0.3, 0.33, 0.333, 0.3333, ...$ converges to $1/3$.

Properties

Limits of sequences behave well with respect to the usual arithmetic operations. If $a_n \to a$ and $b_n \to b$, then $a_n+b_n \to a+b$, $a_nb_n \to ab$ and, if neither b nor any $b_n$ is zero, $a_n/b_n \to a/b$.

For any continuous function f, if $x_n \to x$ then $f(x_n) \to f(x)$. In fact, a function f is continuous if and only if it preserves the limits of sequences.

Infinite limits

The terminology and notation of convergence is also used to describe sequences whose terms become very large. A sequence $(x_n)$ is said to tend to infinity, written $x_n \to \infty$ or $\lim_{n\to\infty}x_n = \infty$ if, for every K, there is an N such that, for every $n \geq N$, $x_n > K$; that is, the sequence terms are eventually larger that any fixed K. Similarly, $x_n \to -\infty$ if, for every K, there is an N such that, for every $n \geq N$, $x_n < K$.

Hyperreal definition

The definition of the limit using the hyperreal numbers formalizes the intuition that for a "very large" value of the index, the corresponding term is "very close" to the limit. More precisely, a sequence x_n tends to L if for every infinite hypernatural H, the term x_H is infinitely close to L, i.e., the difference x_H - L is infinitesimal. Equivalently, L is the standard part of x_H

$L = {\rm st}(x_H)\,$.

Thus, the limit can be defined by the formula

$\lim_{n \to \infty} x_n= {\rm st}(x_H),$

where the limit exists if and only if the righthand side is independent of the choice of an infinite H.

Metric spaces

Definition

A point x of the metric space (X, d) is the limit of the sequence (x_n) if, for all ε > 0, there is an N such that, for every $n \geq N$, $d(x_n, x) < \epsilon$. This coincides with the definition given for real numbers when $X = \mathbb{R}$ and $d(x, y) = |x-y|$.

Properties

For any continuous function f, if $x_n \to x$ then $f(x_n) \to f(x)$. In fact, a function f is continuous if and only if it preserves the limits of sequences.

Limits of sequences are unique when they exist, as distinct points are separated by some positive distance, so for $\epsilon$ less that half this distance, sequence terms cannot be within a distance $\epsilon$ of both points.

Topological spaces

Definition

A point x of the topological space (X, τ) is the limit of the sequence (x_n) if, for every neighbourhood U of x, there is an N such that, for every $n \geq N$, $x_n \in U$. This coincides with the definition given for metric spaces if (X,d) is a metric space and $\tau$ is the topology generated by d.

The limit of a sequence of points $\left(x_n:n\in \mathbb{N}\right)\;$ in a topological space T is a special case of the limit of a function: the domain is $\mathbb{N}$ in the space $\mathbb{N} \cup \lbrace +\infty \rbrace$ with the induced topology of the affinely extended real number system, the range is T, and the function argument n tends to +∞, which in this space is a limit point of $\mathbb{N}$.

Properties

If X is a Hausdorff space then limits of sequences are unique where they exist.

History

The Greek philosopher Zeno of Elea is famous for formulating paradoxes that involve limiting processes.

Leucippus, Democritus, Antiphon, Eudoxus and Archimedes developed the method of exhaustion, which uses an infinite sequence of approximations to determine an area or a volume. Archimedes succeeded in summing what is now called a geometric series.

Newton dealt with series in his works on Analysis with infinite series (written in 1669, circulated in manuscript, published in 1711), Method of fluxions and infinite series (written in 1671, published in English translation in 1736, Latin original published much later) and Tractatus de Quadratura Curvarum (written in 1693, published in 1704 as an Appendix to his Optiks). In the latter work, Newton considers the binomial expansion of (x+o)ⁿ which he then linearizes by taking limits (letting o→0).

In the 18th century, mathematicians like Euler succeeded in summing some divergent series by stopping at the right moment; they did not much care whether a limit existed, as long as it could be calculated. At the end of the century, Lagrange in his Théorie des fonctions analytiques (1797) opined that the lack of rigour precluded further development in calculus. Gauss in his etude of hypergeometric series (1813) for the first time rigorously investigated under which conditions a series converged to a limit.

The modern definition of a limit (for any ε there exists an index N so that ...) was given by Bernhard Bolzano (Der binomische Lehrsatz, Prague 1816, little noticed at the time) and by Weierstrass in the 1870s.

See also

• Limit of a function
• Limit of a net - a net is a topological generalization of a sequence
• Modes of convergence

References

• Frank Morley and James Harkness A treatise on the theory of functions (New York: Macmillan, 1893)

External links

• Examples of sequences
• A history of the calculus, including limits