Limit of a sequence

In mathematics, the limit of a sequence is the value that the terms of a sequence "tend to".[1] If such a limit exists, the sequence is called convergent. A sequence that does not converge is said to be divergent.[2] The limit of a sequence is said to be the fundamental notion on which the whole of analysis ultimately rests.[1]

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. ${\displaystyle 2\pi r}$. 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 ${\displaystyle n}$ becomes larger and larger, the value ${\displaystyle n\cdot \sin \left({\tfrac {1}{n}}\right)}$ becomes arbitrarily close to ${\displaystyle 1}$. We say that "the limit of the sequence ${\displaystyle n\cdot \sin \left({\tfrac {1}{n}}\right)}$ equals ${\displaystyle 1}$."

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

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 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 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 Karl Weierstrass in the 1870s.

Real numbers

The plot of a convergent sequence {a_n} is shown in blue. Visually we can see that the sequence is converging to the limit 0 as n increases.

In the real numbers, a number ${\displaystyle L}$ is the limit of the sequence ${\displaystyle (x_{n})}$ if the numbers in the sequence become closer and closer to ${\displaystyle L}$ and not to any other number.

Examples

See also: List of limits

• If ${\displaystyle x_{n}=c}$ for constant c, then ${\displaystyle x_{n}\to c}$.[proof 1]
• If ${\displaystyle x_{n}={\frac {1}{n}}}$, then ${\displaystyle x_{n}\to 0}$.[proof 2]
• If ${\displaystyle x_{n}=1/n}$ when ${\displaystyle n}$ is even, and ${\displaystyle x_{n}={\frac {1}{n^{2}}}}$ when ${\displaystyle n}$ is odd, then ${\displaystyle x_{n}\to 0}$. (The fact that ${\displaystyle x_{n+1}>x_{n}}$ whenever ${\displaystyle n}$ 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 ${\displaystyle 0.3,0.33,0.333,0.3333,...}$ converges to ${\displaystyle 1/3}$. Note that the decimal representation ${\displaystyle 0.3333...}$ is the limit of the previous sequence, defined by

${\displaystyle 0.3333...\triangleq \lim _{n\to \infty }\sum _{i=1}^{n}{\frac {3}{10^{i}}}}$.

• Finding the limit of a sequence is not always obvious. Two examples are ${\displaystyle \lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n}}$ (the limit of which is the number e) and the Arithmetic–geometric mean. The squeeze theorem is often useful in such cases.

Formal definition

We call ${\displaystyle x}$ the limit of the sequence ${\displaystyle (x_{n})}$ if the following condition holds:

• For each real number ${\displaystyle \epsilon >0}$, there exists a natural number ${\displaystyle N}$ such that, for every natural number ${\displaystyle n\geq N}$, we have ${\displaystyle |x_{n}-x|<\epsilon }$.

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

Symbolically, this is:

• ${\displaystyle \forall \varepsilon >0(\exists N\in \mathbb {N} (\forall n\in \mathbb {N} (n\geq N\implies |x_{n}-x|<\varepsilon ))).}$

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

Illustration

• 
  Example of a sequence which converges to the limit ${\displaystyle a}$.
• 
  Regardless which ${\displaystyle \varepsilon >0}$ we have, there is an index ${\displaystyle N_{0}}$, so that the sequence lies afterwards completely in the epsilon tube ${\displaystyle (a-\varepsilon ,a+\varepsilon )}$.
• 
  There is also for a smaller ${\displaystyle \epsilon _{1}>0}$ an index ${\displaystyle N_{1}}$, so that the sequence is afterwards inside the epsilon tube ${\displaystyle (a-\varepsilon _{1},a+\varepsilon _{1})}$.
• 
  For each ${\displaystyle \varepsilon >0}$ there are only finitely many sequence members outside the epsilon tube.

Properties

Limits of sequences behave well with respect to the usual arithmetic operations. If ${\displaystyle a_{n}\to a}$ and ${\displaystyle b_{n}\to b}$, then ${\displaystyle a_{n}+b_{n}\to a+b}$, ${\displaystyle a_{n}\cdot b_{n}\to ab}$ and, if neither b nor any ${\displaystyle b_{n}}$ is zero, ${\displaystyle {\frac {a_{n}}{b_{n}}}\to {\frac {a}{b}}}$.

For any continuous function f, if ${\displaystyle x_{n}\to x}$ then ${\displaystyle f(x_{n})\to f(x)}$. 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.
• ${\displaystyle \lim _{n\to \infty }(a_{n}\pm b_{n})=\lim _{n\to \infty }a_{n}\pm \lim _{n\to \infty }b_{n}}$
• ${\displaystyle \lim _{n\to \infty }ca_{n}=c\cdot \lim _{n\to \infty }a_{n}}$
• ${\displaystyle \lim _{n\to \infty }(a_{n}\cdot b_{n})=(\lim _{n\to \infty }a_{n})\cdot (\lim _{n\to \infty }b_{n})}$
• ${\displaystyle \lim _{n\to \infty }\left({\frac {a_{n}}{b_{n}}}\right)={\frac {\lim \limits _{n\to \infty }a_{n}}{\lim \limits _{n\to \infty }b_{n}}}}$ provided ${\displaystyle \lim _{n\to \infty }b_{n}\neq 0}$
• ${\displaystyle \lim _{n\to \infty }a_{n}^{p}=\left[\lim _{n\to \infty }a_{n}\right]^{p}}$
• If ${\displaystyle a_{n}\leq b_{n}}$ for all ${\displaystyle n}$ greater than some ${\displaystyle N}$, then ${\displaystyle \lim _{n\to \infty }a_{n}\leq \lim _{n\to \infty }b_{n}}$
• (Squeeze theorem) If ${\displaystyle a_{n}\leq c_{n}\leq b_{n}}$ for all ${\displaystyle n>N}$, and ${\displaystyle \lim _{n\to \infty }a_{n}=\lim _{n\to \infty }b_{n}=L}$,   then ${\displaystyle \lim _{n\to \infty }c_{n}=L}$.
• 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. Once proven that ${\displaystyle {\frac {1}{n}}\to 0}$ it becomes easy to show that ${\displaystyle {\frac {a}{b+{\frac {c}{n}}}}\to {\frac {a}{b}}}$, (${\displaystyle b\neq 0}$), using the properties above.

Infinite limits

A sequence ${\displaystyle (x_{n})}$ is said to tend to infinity, written ${\displaystyle x_{n}\to \infty }$ or ${\displaystyle \lim _{n\to \infty }x_{n}=\infty }$ if, for every K, there is an N such that, for every ${\displaystyle n\geq N}$, ${\displaystyle x_{n}>K}$; that is, the sequence terms are eventually larger than any fixed K. Similarly, ${\displaystyle x_{n}\to -\infty }$ if, for every K, there is an N such that, for every ${\displaystyle n\geq N}$, ${\displaystyle x_{n}<K}$. If a sequence tends to infinity, or to minus infinity, then it is divergent (however, a divergent sequence need not tend to plus or minus infinity: take for example ${\displaystyle x_{n}=(-1)^{n}}$).

Metric spaces

Definition

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

Properties

For any continuous function f, if ${\displaystyle x_{n}\to x}$ then ${\displaystyle f(x_{n})\to f(x)}$. 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 ${\displaystyle \epsilon }$ less than half this distance, sequence terms cannot be within a distance ${\displaystyle \epsilon }$ of both points.

Topological spaces

Definition

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

A limit of a sequence of points ${\displaystyle \left(x_{n}:n\in \mathbb {N} \right)\;}$ in a topological space T is a special case of a limit of a function: the domain is ${\displaystyle \mathbb {N} }$ in the space ${\displaystyle \mathbb {N} \cup \lbrace +\infty \rbrace }$ 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 ${\displaystyle \mathbb {N} }$.

Properties

If X is a Hausdorff space then limits of sequences are unique where they exist. Note that this need not be the case in general; in particular, if two points x and y are topologically indistinguishable, any sequence that converges to x must converge to y and vice versa.

Cauchy sequences

Main article: Cauchy sequence

The plot of a Cauchy sequence (x_n), shown in blue, as x_n 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 numbers

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 ${\displaystyle (x_{n})}$ 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

${\displaystyle L={\rm {st}}(x_{H}).\,}$

Thus, the limit can be defined by the formula

${\displaystyle \lim _{n\to \infty }x_{n}={\rm {st}}(x_{H}),}$

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

See also

• Limit of a net — A net is a topological generalization of a sequence.
• Modes of convergence
• Shift rule

Notes

1. Courant (1961), p. 29.
2. Courant (1961), p. 39.

Proofs

1. Proof: choose ${\displaystyle N=1}$. For every ${\displaystyle n\geq N}$, ${\displaystyle |x_{n}-c|=0<\epsilon }$
2. Proof: choose ${\displaystyle N=\left\lfloor {\frac {1}{\epsilon }}\right\rfloor }$ + 1 (the floor function). For every ${\displaystyle n\geq N}$, ${\displaystyle |x_{n}-0|\leq x_{N}={\frac {1}{\lfloor 1/\epsilon \rfloor +1}}<\epsilon }$.

References

• 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 links

• "Limit", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• A history of the calculus, including limits