Limit of a sequence


In mathematics, the limit of a sequence is the value that the terms of a sequence "tend to", and is often denoted using the symbol (e.g., ).[1] If such a limit exists, the sequence is called convergent.[2] A sequence that does not converge is said to be divergent.[3] The limit of a sequence is said to be the fundamental notion on which the whole of mathematical analysis ultimately rests.[1]

diagram of a hexagon and pentagon circumscribed outside a circle
The sequence given by the perimeters of regular n-sided polygons that circumscribe the unit circle has a limit equal to the perimeter of the circle, i.e. The corresponding sequence for inscribed polygons has the same limit.
n n sin(1/n)
1 0.841471
2 0.958851
10 0.998334
100 0.999983

As the positive integer becomes larger and larger, the value becomes arbitrarily close to We say that "the limit of the sequence equals "

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


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.

Grégoire de Saint-Vincent gave the first definition of limit (terminus) of a geometric series in his work Opus Geometricum (1647): "The terminus of a progression is the end of the series, which none progression can reach, even not if she is continued in infinity, but which she can approach nearer than a given segment."[4]

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 the limit as o tends to 0.

In the 18th century, mathematicians such as 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 the conditions under which 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, which was little noticed at the time), and by Karl Weierstrass in the 1870s.

Real numbersEdit

The plot of a convergent sequence {an} is shown in blue. Here, one can see that the sequence is converging to the limit 0 as n increases.

In the real numbers, a number   is the limit of the sequence   if the numbers in the sequence become closer and closer to  —and not to any other number.


  • If   for constant c, then  [proof 1][5]
  • If  , then  .[proof 2][5]
  • If   when   is even, and   when   is odd, then   (The fact that   whenever   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   converges to   Note that the decimal representation   is the limit of the previous sequence, defined by
  • Finding the limit of a sequence is not always obvious. Two examples are   (the limit of which is the number e) and the Arithmetic–geometric mean. The squeeze theorem is often useful in the establishment of such limits.

Formal definitionEdit

We call   the limit of the sequence   if the following condition holds:

  • For each real number   there exists a natural number   such that, for every natural number   we have  [6]

In other words, for every measure of closeness   the sequence's terms are eventually that close to the limit. The sequence   is said to converge to or tend to the limit   written   or  

Symbolically, this is:


If a sequence   converges to some limit   then it is convergent and   is the only limit; otherwise   is divergent. A sequence that has zero as its limit is sometimes called a null sequence.


Properties (real numbers)Edit

Limits of sequences behave well with respect to the usual arithmetic operations. If   and   then     and, if neither b nor any   is zero,  [5]

For any continuous function f, if   then   In fact, any real-valued function f is continuous if and only if it preserves the limits of sequences (though this is not necessarily true when using more general notions of continuity).

Some other important properties of limits of real sequences include the following (provided, in each equation below, that the limits on the right exist).

  • The limit of a sequence is unique.[5]
  •  [5]
  •  [5]
  •  [5]
  •   provided  [5]
  • If   for all   greater than some   then  
  • (Squeeze theorem) If   for all   and   then  
  • If a sequence is bounded and monotonic, then it is convergent.
  • A sequence is convergent if and only if every subsequence is convergent.
  • If every subsequence of a sequence has its own subsequence which converges to the same point, then the original sequence converges to that point.

These properties are extensively used to prove limits, without the need to directly use the cumbersome formal definition. For example. once it is proven that   it becomes easy to show—using the properties above—that   (assuming that  ).

Infinite limitsEdit

A sequence   is said to tend to infinity, written   or   if for every K, there is an N such that for every    ; that is, the sequence terms are eventually larger than any fixed K.

Similarly,   if for every K, there is an N such that for every     If a sequence tends to infinity or minus infinity, then it is divergent. However, a divergent sequence need not tend to plus or minus infinity, and the sequence   provides one such example.

Metric spacesEdit


A point   of the metric space   is the limit of the sequence   if for all   there is an   such that, for every     This coincides with the definition given for real numbers when   and  

Properties (metric spaces)Edit

For any continuous function f, if   then   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   less than half this distance, sequence terms cannot be within a distance   of both points.

Topological spacesEdit


A point   of the topological space   is a limit or limit point[7][8] of the sequence   if for every neighbourhood   of   there exists some   such that for every    [9] This coincides with the definition given for metric spaces, if   is a metric space and   is the topology generated by  

A limit of a sequence of points   in a topological space   is a special case of a limit of a function: the domain is   in the space   with the induced topology of the affinely extended real number system, the range is   and the function argument   tends to   which in this space is a limit point of  

Properties (topological spaces)Edit

In a Hausdorff space, limits of sequences are unique whenever they exist. Note that this need not be the case in non-Hausdorff spaces; in particular, if two points   and   are topologically indistinguishable, then any sequence that converges to   must converge to   and vice versa.

Cauchy sequencesEdit

The plot of a Cauchy sequence (xn), shown in blue, as   versus n. Visually, we see that the sequence appears to be converging to a limit point as the terms in the sequence become closer together as n increases. In the real numbers every Cauchy sequence converges to some limit.

A Cauchy sequence is a sequence whose terms ultimately become arbitrarily close together, after sufficiently many initial terms have been discarded. The notion of a Cauchy sequence is important in the study of sequences in metric spaces, and, in particular, in real analysis. One particularly important result in real analysis is the Cauchy criterion for convergence of sequences: a sequence of real numbers is convergent if and only if it is a Cauchy sequence. This remains true in other complete metric spaces.

Definition in hyperreal numbersEdit

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 real sequence   tends to L if for every infinite hypernatural H, the term   is infinitely close to L (i.e., the difference   is infinitesimal). Equivalently, L is the standard part of  


Thus, the limit can be defined by the formula

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

See alsoEdit


  1. ^ a b Courant (1961), p. 29.
  2. ^ Weisstein, Eric W. "Convergent Sequence". Retrieved 2020-08-18.
  3. ^ Courant (1961), p. 39.
  4. ^ Van Looy, H. (1984). A chronology and historical analysis of the mathematical manuscripts of Gregorius a Sancto Vincentio (1584–1667). Historia Mathematica, 11(1), 57-75.
  5. ^ a b c d e f g h "Limits of Sequences | Brilliant Math & Science Wiki". Retrieved 2020-08-18.
  6. ^ Weisstein, Eric W. "Limit". Retrieved 2020-08-18.
  7. ^ Dugundji 1966, pp. 209–210.
  8. ^ Császár 1978, p. 61.
  9. ^ Zeidler, Eberhard (1995). Applied functional analysis : main principles and their applications (1 ed.). New York: Springer-Verlag. p. 29. ISBN 978-0-387-94422-7.


  1. ^ Proof: choose   For every    
  2. ^ Proof: choose   (the floor function). For every    


  • Császár, Ákos (1978). General topology. Translated by Császár, Klára. Bristol England: Adam Hilger Ltd. ISBN 0-85274-275-4. OCLC 4146011.
  • Dugundji, James (1966). Topology. Boston: Allyn and Bacon. ISBN 978-0-697-06889-7. OCLC 395340485.
  • Courant, Richard (1961). "Differential and Integral Calculus Volume I", Blackie & Son, Ltd., Glasgow.
  • Frank Morley and James Harkness A treatise on the theory of functions (New York: Macmillan, 1893)

External linksEdit