Table of contents |
2 Formal definition 3 Examples 4 Properties 5 See Also |
History
See mathematical analysis.
Suppose x1, x2, ... 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
For sequence of real or complex numbers, the metric (distance) between xn and L is the absolute value |xn - L|.
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 infinitely.
A function f : R -> R is continuous if and only if it is compatible with limits in the following sense:
Every convergent sequence is a Cauchy sequence and hence bounded.
If (xn) is a bounded sequence of real numbers which is increasing (i.e. xn ≤ xn+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
These rules are also valid for infinite limits using the rules
Formal definition
if and only if
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.Examples
Properties
A subsequence of the sequence (xn) is a sequence of the form (xa(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.
and
then
and
and
(the latter provided that f2(x) is non-zero in a neighborhood of p and L2 is non-zero as well).
(see extended real number line).