Limit of a sequence
Limit of a sequence is one of the oldest concept in
mathematical analysis. It is the essential tool in calculating
pi and
trigonometric functions.
See mathematical analysis.
Suppose x_{1}, x_{2}, ... is a sequence of elementss in a metric space (M,d).
We say that the real number L is the limit of this sequence and we write
if and only if
 for every ε>0 there exists a natural number n_{0} (which will depend on ε) such that for all n>n_{0} we have d(x_{n},L) < ε.
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. One can show that a convergent sequence has only one limit.
For sequence of real or complex numbers, the metric (distance) between x_{n} and L is the absolute value x_{n}  L.
 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 a^{n} 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 n1<x≤n. Then the limit of the sequence of x_n is just the limit of f(x) at infinitely.
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 is a Cauchy sequence and hence bounded.
If (x_{n}) is a bounded sequence of real numbers which is increasing (i.e. x_{n} ≤ x_{n+1} for all n), then it 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
and
(the latter provided that
f_{2}(x) is nonzero in a
neighborhood of
p and
L_{2} is nonzero as well).
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).