The limit of a sequence is one of the oldest concepts in mathematical analysis. It provides a rigorous definition of the idea of a sequence converging towards a point called the limit. Analysis is the branch of mathematics most explicitly concerned with the notion of a limit, either the limit of a sequence or the limit of a function. ...

Intuitively, suppose we have a sequence of points (i.e. an infinite set of points labelled using the natural numbers) in some sort of mathematical object (for example the real numbers or a vector space) which has a concept of nearness (such as "all points within a given distance of a fixed point"). A point L is the limit of the sequence if for any prescribed nearness, all but a finite number of points in the sequence are that near to L. This may be visualised as a set of spheres of size decreasing to zero, all with the same centre L, and for any one of these balls, only a finite number of points in the sequence being outside the ball. In mathematics, a sequence is a list of objects (or events) arranged in a linear fashion, such that the order of the members is well defined and significant. ... In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ...

Formal definition

• For a sequence of points in a metric space M with distance function d

(such as a sequence of rational numbers, real numbers, complex numbers, points in a normed space, etc.):

If we say L is the limit of the sequence and write

i.e.:if and only if for every real number , there is a natural number N such that for every , we have

• As a generalization of this, for a sequence of points in a topological space T:

If we say L is a limit of this sequence and write

if and only if for every neighborhood S of L there is a natural number N such that for all

If a sequence has a limit, we say the sequence is convergent, and that the sequence converges to the limit. Otherwise, the sequence is divergent. In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ... In mathematics, a rational number (commonly called a fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the length of a vector is intuitive and can be easily extended to any real vector space Rn. ... In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space. ...

Comments

The definition means that eventually all elements of the sequence get as close as we want to the limit. (The condition that the elements become arbitrarily close to all of the following elements does not, in general, imply the sequence has a limit. See Cauchy sequence). In mathematical analysis, a Cauchy sequence, named after Augustin Cauchy, is a sequence whose elements become close as the sequence progresses. ...

Also, a sequence may, in a general topological space, have several different limits, but a convergent sequence has a unique limit if T is a Hausdorff space, for example the (extended) real line, the complex plane, their subsets (R, Q, Z...) and Cartesian products (Rⁿ...). Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ... The extended real number line is obtained from the real number line R by adding two elements: +∞ and −∞ (which are not considered to be real numbers). ... In mathematics, the complex plane is a way of visualising the space of the complex numbers. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, a rational number (commonly called a fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... The integers are commonly denoted by the above symbol. ...

Examples

• The sequence 1/1, 1/2, 1/3, 1/4, ... of real numbers converges with limit 0.
• The sequence 1, -1, 1, -1, 1, ... is divergent.
• The sequence 1/2, 1/2 + 1/4, 1/2 + 1/4 + 1/8, 1/2 + 1/4 + 1/8 + 1/16, ... converges with limit 1. This is an example of an infinite series.
• If a is a real number with absolute value |a| < 1, then the sequence aⁿ has limit 0. If 0 < a ≤ 1, then the sequence a^1/n has limit 1.

Also: In mathematics, a series is often represented as the sum of a sequence of terms. ... In mathematics, the absolute value (or modulus[1]) of a real number is its numerical value without regard to its sign. ...

Properties

Consider the following function: f(x)=x_n if n-1<x≤n. Then the limit of the sequence of x_n is just the limit of f(x) at infinity. In mathematics, the limit of a function is a fundamental concept in mathematical analysis. ...

A function f, defined on a first-countable space, is continuous if and only if it is compatible with limits in that (f(x_n)) converges to f(L) given that (x_n) converges to L, i.e. Partial plot of a function f. ... In topology, a first-countable space is a topological space satisfying the first axiom of countability. Specifically, a space X is said to be first-countable if each point has a countable local base. ... In topology and related areas of mathematics a continuous function is a morphism between topological spaces. ...

implies

Note that this equivalence does not hold in general for spaces which are not first-countable. For functions in the reals, this is often simplified to

f is continuous at x if and only if

A subsequence of the sequence (x_n) is a sequence of the form (x_a(n)) where the a(n) are natural numbers with a(n) < a(n+1) for all n. Intuitively, a subsequence omits some elements of the original sequence. A sequence is convergent if and only if all of its subsequences converge towards the same limit.

Every convergent sequence in a metric space is a Cauchy sequence and hence bounded. A bounded monotonic sequence of real numbers is necessarily convergent: this is sometimes called the fundamental theorem of analysis. More generally, every Cauchy sequence of real numbers has a limit, or short: the real numbers are complete. In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ... In mathematical analysis, a Cauchy sequence, named after Augustin Cauchy, is a sequence whose elements become close as the sequence progresses. ... In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ... In mathematics, functions between ordered sets are monotonic (or monotone) if they preserve the given order. ... In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if it doesnt have any holes, if there arent any points missing. For...

A sequence of real numbers is convergent if and only if its limit inferior and limit superior coincide and are both finite. In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting bounds on the sequence. ... In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting bounds on the sequence. ...

The algebraic operations are everywhere continuous (except for division around zero divisor); thus, given

and

then

and (if L₂ and y_n is non-zero)

These rules are also valid for infinite limits using the rules

• q + ∞ = ∞ for q ≠ -∞
• q × ∞ = ∞ if q > 0
• q × ∞ = -∞ if q < 0
• q / ∞ = 0 if q ≠ ± ∞

(see extended real number line). The extended real number line is obtained from the real number line R by adding two elements: +&#8734; and &#8722;&#8734; (which are not considered to be real numbers). ...

History

The Greek philosopher Zeno of Elea is famous for formulating paradoxes that involve limiting processes. Zeno of Elea (IPA:zɛnoʊ, ɛlɛɑː)(circa 490 BC? – circa 430 BC?) was a pre-Socratic Greek philosopher of southern Italy and a member of the Eleatic School founded by Parmenides. ... This article or section does not adequately cite its references or sources. ...

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. This article is about the philosopher. ... ‎ Democritus (Greek: ) was a pre-Socratic Greek philosopher (born at Abdera in Thrace around 460 BC[1][2]). Democritus was a student of Leucippus and co-originator of the belief that all matter is made up of various imperishable, indivisible elements which he called atomos, from which we get the... This article is about the musical term. ... Eudoxus was the name of two ancient Greeks: Eudoxus of Cnidus (c. ... Archimedes (Greek: c. ... The method of exhaustion is a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape. ...

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). Sir Isaac Newton, (4 January 1643 – 31 March 1727) [ OS: 25 December 1642 – 20 March 1727][1] was an English physicist, mathematician, astronomer, natural philosopher, and alchemist, regarded by many as the greatest figure in the history of science. ...

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. Leonhard Euler is considered by many to be one of the greatest mathematicians of all time A mathematician is the person whose primary area of study and research is the field of mathematics. ... Leonhard Euler (pronounced Oiler; IPA ) (April 15, 1707 – September 7, 1783) was a pioneering Swiss mathematician and physicist, who spent most of his life in Russia and Germany. ... Joseph-Louis Lagrange, comte de lEmpire (January 25, 1736 – April 10, 1813; b. ... Johann Carl Friedrich Gauss or Gauß ( ; Latin: ) (30 April 1777 – 23 February 1855) was a German mathematician and scientist of profound genius who contributed significantly to many fields, including number theory, analysis, differential geometry, geodesy, magnetism, astronomy, and optics. ... In mathematics, a hypergeometric series is a power series in which the ratios of successive coefficients k is a rational function of k. ...

The modern definition of a limit (for any ε there exists an index N so that ...) was given independently by Bernhard Bolzano (Der binomische Lehrsatz, Prag 1816, little noticed at the time) and by Cauchy in his Cours d'analyse (1821). Bernard Bolzano Bernard Placidus Johann Nepomuk Bolzano (October 5, 1781 &#8211; December 18, 1848) was a German-speaking Czech mathematician, theologian, philosopher and logician. ... Augustin Louis Cauchy (August 21, 1789 – May 23, 1857) was a French mathematician. ...

See also mathematical analysis; external link: [1]. Analysis is the branch of mathematics most explicitly concerned with the notion of a limit, either the limit of a sequence or the limit of a function. ...

See also

• Limit of a function
• Net (topology)

In mathematics, the limit of a function is a fundamental concept in mathematical analysis. ... In mathematics the term net has at least two meanings. ...

External links

• Examples of Sequences