Limit of a sequence

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Limit of a sequence is one of the oldest concept in mathematical analysis. It is the essential tool in calculating pi and trigonometric functions.

Table of contents

1 History
2 Formal definition
3 Examples
4 Properties
5 See Also

Formal definition

Suppose x₁, x₂, ... is a sequence of elementss in a topological space T. We say that L∈T is the limit of this sequence and write

if and only if

for every neighbourhood S of L there is an N such that x_n∈S for all n>N.

Intuitively, this means that eventually all elements of the sequence get as close as we want to the limit. Not every sequence has a limit; if it does, we call it convergent, otherwise divergent. In a Hausdorff space, a convergent sequence has a unique limit.

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.

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.

A function f : R -> R is continuous if and only if it is compatible with limits in the following sense:

if (x_n) is any convergent sequence in R with limit L, then the sequence (f(x_n)) converges with limit f(L).

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. More generally, every Cauchy sequence of real numbers has a limit, or short: the real numbers are complete.

A sequence of real numbers is convergent if and only if its limit inferior and limit superior coincide and are both finite.

Taking the limit of sequences is compatible with the algebraic operations: If

and

then

and (if L₂ 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).