Limit superior and limit inferior

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In mathematics, the limit inferior and limit superior (also called infimum limit and supremum limit, or liminf and limsup) of a sequence can be thought of as limiting bounds on the sequence. (See limit of a function.)

The limit inferior of a sequence (x_n) is defined as

$\liminf_{n\rightarrow\infty}x_n=\sup_{n\geq 0}\,\inf_{k\geq n}x_k=\sup\{\,\inf\{\,x_k:k\geq n\,\}:n\geq 0\,\}.$

$\liminf_{n\rightarrow\infty}x_n=\lim_{n\rightarrow\infty}\Big(\inf_{m\geq n}x_m\Big).$

Similarly, the limit superior of (x_n) is defined as

$\limsup_{n\rightarrow\infty}x_n=\inf_{n\geq 0}\,\sup_{k\geq n}x_k=\inf\{\,\sup\{\,x_k:k\geq n\,\}:n\geq 0\,\}.$

$\limsup_{n\rightarrow\infty}x_n=\lim_{n\rightarrow\infty}\Big(\sup_{m\geq n}x_m\Big).$

These definitions make sense in any partially ordered set, provided the suprema and infima exist. In a complete lattice, the suprema and infima always exist, and so in this case every sequence has a limit superior and a limit inferior.

Whenever the ordinary limit exists, the limit inferior and limit superior are both equal to it; therefore, each can be considered a generalization of the ordinary limit which is primarily interesting in cases where the limit does not exist. Whenever lim inf x_n and lim sup x_n both exist, we have

$\liminf_{n\rightarrow\infty}x_n\leq\limsup_{n\rightarrow\infty}x_n.$

Limits inferior/superior are related to big-O notation in that they bound a sequence only "in the limit"; the sequence may exceed the bound. However, with big-O notation the sequence can only exceed the bound in a finite prefix of the sequence, whereas the limit superior of a sequence like e^-n may actually be less than all elements of the sequence. The only promise made is that some tail of the sequence can be bounded by the limit superior (inferior) plus (minus) an arbitrarily small positive constant.

1 Another definition
2 Sequences of real numbers
3 Functions from metric and topological spaces to the real numbers
4 Sequences of sets

[edit] Another definition

Limit superior of $x n$ is a number $b$ that, for any positive number $\varepsilon$ , $b+\varepsilon<x_n$ holds for finitely many $n$ and $b-\varepsilon<x_n$ for infinitely many $n$ .

Limit inferior of $x n$ is a number $b$ that, for any positive number $\varepsilon$ , $b-\varepsilon>x_n$ holds for finitely many $n$ and $b+\varepsilon>x_n$ for infinitely many $n$ .

[edit] Sequences of real numbers

In calculus, the case of sequences in R (the real numbers) is important. R itself is not a complete lattice, but positive and negative infinities can be added to give the complete totally ordered set [-∞,∞]. Then (x_n) in [-∞,∞] converges if and only if

$\liminf_{n\to\infty} x_n = \limsup_{n\to\infty} x_n,$

in which case

$\lim_{n\to\infty} x_n$

is equal to their common value. (Note that when working just in R, convergence to -∞ or ∞ would not be considered as convergence.) Since the limit inferior is at most the limit superior, the condition

$\liminf_{n\to\infty} x_n = \infty$

implies that

$\lim_{n\to\infty} x_n = \infty,$

and the condition

$\limsup_{n\to\infty} x_n = - \infty$

implies that

$\lim_{n\to\infty} x_n = - \infty$ .

As an example, consider the sequence given by x_n = sin(n). Using the fact that pi is irrational, one can show that

$\liminf_{n\to\infty} x_n = -1$

and

$\limsup_{n\to\infty} x_n = +1.$

(This is because the sequence {1,2,3,...} is equidistributed mod 2π, a consequence of the Equidistribution theorem.)

$I = \liminf_{n\to\infty} x_n$

and

$S = \limsup_{n\to\infty} x_n,$

then the interval [I, S] need not contain any of the numbers x_n, but every slight enlargement [I − ε, S + ε] (for arbitrarily small ε > 0) will contain x_n for all but finitely many indices n. In fact, the interval [I, S] is the smallest closed interval with this property.

An example from number theory is

$\liminf_{n\to\infty}(p_{n+1}-p_n),$

where p_n is the n-th prime number. The value of this limit inferior is conjectured to be 2 - this is the twin prime conjecture - but as yet has not even been proved finite.

[edit] Functions from metric and topological spaces to the real numbers

There is a notion of lim sup and lim inf for real-valued functions defined on a metric space whose relationship to limits of real-valued functions mirrors that of the relation between the lim sup, lim inf, and the limit of a real sequence. Given a metric space X, a subspace E contained in X, and f : E → R we define, for a any point in the closure of E:

$\limsup_{x \to a} f(x) = \lim_{\varepsilon \to 0} \sup \{ f(x) : x \in E \cap B(a;\varepsilon) \}$

and

$\liminf_{x \to a} f(x) = \lim_{\varepsilon \to 0} \inf \{ f(x) : x \in E \cap B(a;\varepsilon) \}$

where B(a,ε) denotes the metric ball of radius ε about a. As in the case for sequences, these are always well-defined if we allow the values +∞ and -∞, and if both are equal then the limit exists and is equal to their common value (again possibly including the infinities). For example, given f(x) = sin(1/x), we have lim sup_x→0 f(x) = 1 and lim inf_x→0 f(x) = -1. The difference between the two is a rough measure of how "wildly" the function oscillates, and in observation of this fact, it is called the oscillation of f at a. This idea of oscillation is sufficient to, for example, characterize Riemann-integrable functions as continuous except on a set of measure zero [1]. Note that points of nonzero oscillation i.e. points at which f is "badly behaved" are discontinuities which, unless they make up a set of zero, are confined to a negligible set.

Note that as ε shrinks, the supremum of the function over the ball is monotone decreasing, so we have

$\limsup_{x\to a} f(x) = \inf_{\varepsilon > 0} (\sup \{ f(x) : x \in E \cap B(a;\varepsilon) \})$

and similarly

$\liminf_{x\to a} f(x) = \sup_{\varepsilon > 0}(\inf \{ f(x) : x \in E \cap B(a;\varepsilon) \}).$

This finally motivates the definitions for general topological spaces, for X, E and a as before, but now X only a topological space, we replace balls with neighborhoods:

$\limsup_{x\to a} f(x) = \inf \{ \sup \{ f(x) : x \in E \cap U \} : U\ \mathrm{open}, a \in U \}$

$\liminf_{x\to a} f(x) = \sup \{ \inf \{ f(x) : x \in E \cap U \} : U\ \mathrm{open}, a \in U \}$

(there is a way to write the formula using a lim using nets and the neighborhood filter). This version is often useful in discussions of semi-continuity which crop up in analysis quite often. An interesting note is that this version subsumes the sequential version by considering sequences as functions from the natural numbers as a topological subspace of the extended real line, into the space (the closure of N in [-∞, ∞] is N ∪ {∞}.)

[edit] Sequences of sets

The power set P(X) of a set X is a complete lattice, and it is sometimes useful to consider limits superior and inferior of sequences in P(X), that is, sequences of subsets of X. If X_n is such a sequence, then an element a of X belongs to lim inf X_n if and only if there exists a natural number n₀ such that a is in X_n for all n > n₀. The element a belongs to lim sup X_n if and only if for every natural number n₀ there exists an index n > n₀ such that a is in X_n. In other words, lim sup X_n consists of those elements which are in X_n for infinitely many n, while lim inf X_n consists of those elements which are in X_n for all but finitely many n.

Using the standard parlance of set theory, the infimum of a sequence of sets is the countable intersection of the sets, the largest set included in all of the sets:

$\inf\left\{\,x_n : n=1,2,3,\dots\,\right\}={\bigcap_{n=1}^\infty}x_n.$

The sequence of I_n, n=1,2,3,..., where I_n is the infimum of set n, is non-decreasing, because I_n⊂I_n+1. Therefore, the countable union of infimum from 1 to n is equal to the nth infimum. Taking this sequence of sets to the limit:

$\liminf_{n\rightarrow\infty}x_n={\bigcup_{n=1}^\infty}\left({\bigcap_{m=n}^\infty}x_m\right).$

The limsup can be defined in a dual fashion. The supremum of a sequence of sets is the smallest set containing all the sets, i.e., the countable union of the sets.

$\sup\left\{\,x_n : n=1,2,3,\dots\,\right\}={\bigcup_{n=1}^\infty}x_n.$