Limit of a sequence







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



In mathematics, the limit of a sequence is the value that the terms of a sequence "tend to". If such a limit exists, the sequence is called convergent. A sequence that does not converge is said to be divergent. The limit of a sequence is said to be the fundamental notion on which the whole of analysis ultimately rests.
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, Method of fluxions and infinite series and Tractatus de Quadratura Curvarum. In the latter work, Newton considers the binomial expansion of n which he then linearizes by taking limits.
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 opined that the lack of rigour precluded further development in calculus. Gauss in his etude of hypergeometric series for the first time rigorously investigated under which conditions a series converged to a limit.
The modern definition of a limit was given by Bernhard Bolzano and by Karl Weierstrass in the 1870s.

Real numbers

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.

Examples

We call the limit of the sequence if the following condition holds:
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; otherwise it is divergent.

Illustration

Properties

Limits of sequences behave well with respect to the usual arithmetic operations. If and, then, and, if neither b nor any is zero,.
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.
Some other important properties of limits of real sequences include the following.
These properties are extensively used to prove limits without the need to directly use the cumbersome formal definition. Once proven that
it becomes easy to show that,, using the properties above.

Infinite limits

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 to minus infinity, then it is divergent .

Metric spaces

Definition

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

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 spaces

Definition

A point x of the topological space is a limit of the sequence if, for every neighbourhood U of x, there is an N such that, for every,. This coincides with the definition given for metric spaces if is a metric space and is the topology generated by d.
A limit of a sequence of points in a topological space T 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 T, and the function argument n tends to +∞, which in this space is a limit point of.

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

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

Proofs