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 (i.e., eventual and extreme) bounds on the sequence. The limit inferior and limit superior of a function can be thought of in a similar fashion (see limit of a function.) The limit inferior and limit superior of a set are the infimum and supremum of the set's limit points respectively.

An illustration of limit superior and limit inferior. The sequence x_n is shown in blue. The two red curves approach the limit superior and limit inferior of x_n, shown as solid red lines to the right.

1 Definition for sequences
2 The case of sequences of real numbers
- 2.1 Interpretation
- 2.2 Properties
3 Real-valued functions
4 Functions from metric spaces to metric spaces
5 Sequences of sets
6 Generalized definitions
- 6.1 Definition for a set
- 6.2 Definition for filter bases
  - 6.2.1 Specialization for sequences and nets
7 See also
8 References

[edit] Definition for sequences

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

$\liminf_{n\rightarrow\infty}x_n=\sup_{n\geq 0}\,\inf_{m\geq n}x_m=\sup\{\,\inf\{\,x_m:m\geq n\,\}:n\geq 0\,\}$

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

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

$\limsup_{n\rightarrow\infty}x_n=\inf_{n\geq 0}\,\sup_{m\geq n}x_m=\inf\{\,\sup\{\,x_m:m\geq n\,\}:n\geq 0\,\}$

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

If the terms in the sequence are real numbers, the limit superior and limit inferior always exist, as real numbers or ±∞. More generally, these definitions make sense in any partially ordered set, provided the suprema and infima exist, such as in a complete lattice.

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.

The limit superior and limit inferior of a sequence are a special case of those of a function (see below).

[edit] The case of sequences of real numbers

In mathematical analysis, limit superior and limit inferior are important tools for studying sequences of real numbers. In order to deal with the difficulties arising from the fact that the supremum and infimum of a set of real numbers may not exist (the reals are not a complete lattice), it is convenient to consider sequences in the affinely extended real number system: we add the positive and negative infinities to the real line to give the complete totally ordered set [-∞,∞], which is a complete lattice.

[edit] Interpretation

Consider a sequence $(x n)$ consisting of real numbers. Assume that the limit superior and limit inferior are real numbers (so, not infinite).

The limit superior of $x n$ is the smallest real number $b$ such that, for any positive real number $\varepsilon$ , there exists a natural number $N$ such that $x_n<b+\varepsilon$ for all $n > N$ . In other words, any number larger than the limit superior is an eventual upper bound for the sequence. Only a finite number of elements of the sequence are greater than $b+\varepsilon$ .
The limit inferior of $x n$ is the largest real number $b$ that, for any positive real number $\varepsilon$ , there exists a natural number $N$ such that $x_n>b-\varepsilon$ for all $n > N$ . In other words, any number below the limit inferior is an eventual lower bound for the sequence. Only a finite number of elements of the sequence are less than $b-\varepsilon$ .

[edit] Properties

The relationship of limit inferior and limit superior for sequences of real numbers is as follows

$-\liminf_{n\to\infty} x_n = \limsup_{n\to\infty} (-x_n).$

As mentioned earlier, it is convenient to extend R to [-∞,∞]. 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. We can formalize this property like this. If there exists a $\Lambda\in\mathbb{R}$ so that

$\Lambda<\limsup_{n\to\infty}x_n$

then there exists a subsequence $x_{k_n}$ of $x n$ for which we have that

$x_{k_n}>\Lambda\quad \forall n$

In the same way we get for the liminf: If

$\liminf_{n\to\infty}x_n<\lambda$

there exists exists a subsequence $x_{k_n}$ of $x n$ for which we have that

$x_{k_n}<\lambda\quad\forall n$ .

On the other hand we have that if

$\limsup_{n\to\infty}x_n<\Lambda$

there exists a $n_0\in\mathbb{N}$ so that

$x_n<\Lambda\quad\forall n\geq n_0$

Similarly we get for the liminf that if there exists a $\lambda\in \mathbb{R}$ so that

$\lambda<\liminf_{n\to\infty}x_n$

there exists a $n_0\in\mathbb{N}$ so that

$x_n>\lambda\quad\forall n\geq n_0$

To recapitulate:

If $Λ$ is greater than the limit superior, there are at most finitely many $x n$ greater than $Λ$ ; if it is less, there are infinitely many.
If $λ$ is less than the limit inferior, there are at most finitely many $x n$ less than $λ$ ; if it is greater, there are infinitely many.

In general we have that

$\inf_n x_n \leq \liminf_{n \to \infty} x_n \leq \limsup_{n \to \infty} x_n \leq \sup_n x_n$

The liminf and limsup of a sequence are respectively the smallest and greatest cluster points.

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. The corresponding limit superior is $+\infty$ , because there are arbitrary gaps between consecutive primes.

[edit] Real-valued functions

Assume that a function is defined from a subset of the real numbers to the real numbers. As in the case for sequences, the limit inferior and limit superior are always well-defined if we allow the values +∞ and -∞; in fact, if both agree 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.

[edit] Functions from metric spaces to metric spaces

There is a notion of lim sup and lim inf for 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. Take metric spaces X and Y, a subspace E contained in X, and a function f : E → Y. The space Y should also be an ordered set, so that the notions of supremum and infimum make sense. Define, for any limit point a of E,

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

and

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

where B(a;ε) denotes the metric ball of radius ε about a. (Note that the right hand side is an ordinary limit of a supremum or infimum, not itself a lim-sup or lim-inf.)

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

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

and similarly

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

This finally motivates the definitions for general topological spaces. Take X, Y, E and a as before, but now let X and Y both be topological spaces. In this case, we replace metric balls with neighborhoods:

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

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

(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.

As an example, consider the sequence

$\{0\},\{1\},\{0\},\{1\},\{0\},\{1\},\dots$

whose superior limit is {0,1} but whose inferior limit is empty.

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 infima

$I_{n = 1,2,3,\dots} = \inf \left\{ X_m : m=n,\dots \right\} = {\bigcap_{m=n}^\infty}X_m$

is non-decreasing, because it is the intersection of ever fewer sets. Therefore, $I_n \subset I_{n + 1}$ and the countable union of infima from 1 to n is equal to the n^th 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.

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

The limsup is the countable intersection of this non-increasing (each supremum is a subset of the previous supremum) sequence of sets.

$\limsup_{n\rightarrow\infty}X_n={\bigcap_{n=1}^\infty}\left({\bigcup_{m=n}^\infty}X_m\right)$

See Borel-Cantelli lemma for an example.

[edit] Generalized definitions

The above definitions are inadequate for many technical applications. In fact, the definitions above are specializations of the following definitions.

[edit] Definition for a set

The limit inferior of a set X is the infimum of all of the limit points of the set. That is,

$\liminf X = \inf \{ x \in X : x \text{ is a limit point of } X \}\,$

Similarly, the limit superior of a set X is the supremum of all of the limit points of the set. That is,

$\limsup X = \sup \{ x \in X : x \text{ is a limit point of } X \}\,$

Note that the set X needs to be defined as a subset of a partially ordered set that is also a topological space in order for these definitions to make sense. Moreover, it has to be a complete lattice, so that the suprema and infima always exist. In that case every set has a limit superior and a limit inferior. Also note that neither the limit inferior nor the limit superior of a set must be an element of the set.

[edit] Definition for filter bases

Take a topological space X and a filter base B in that space. The set of all cluster points for that filter base is given by

$\bigcap \{ \overline{B}_0 : B_0 \in B \}$

where $\overline{B}_0$ is the closure of $B 0$ . This is clearly a closed set and is similar to the set of limit points of a set. Assume that X is also a partially ordered set. The limit superior of the filter base B is defined as

$\limsup B = \sup \bigcap \{ \overline{B}_0 : B_0 \in B \}$

when that supremum exists. When X has a total order, is a complete lattice and has the order topology,

$\limsup B = \inf\{ \sup B_0 : B_0 \in B \}$

Proof: Similarly, the limit inferior of the filter base B is defined as

$\liminf B = \inf \bigcap \{ \overline{B}_0 : B_0 \in B \}$

when that infimum exists; if X is totally ordered, is a complete lattice, and has the order topology, then

$\liminf B = \sup\{ \inf B_0 : B_0 \in B \}$

If the limit inferior and limit superior agree, then there must be exactly one cluster point and the limit of the filter base is equal to this unique cluster point.

[edit] Specialization for sequences and nets

Note that filter bases are generalizations of nets, which are generalizations of sequences. Therefore, these definitions give the limit inferior and limit superior of any net (and thus any sequence) as well. For example, take topological space $X$ and the net $(x_\alpha)_{\alpha \in A}$ , where $(A,{\leq})$ is a directed set and $x_\alpha \in X$ for all $\alpha \in A$ . The filter base ("of tails") generated by this net is $B$ defined by

$B \triangleq \{ \{ x_\alpha : \alpha_0 \leq \alpha \} : \alpha_0 \in A \}.\,$

Therefore, the limit inferior and limit superior of the net are equal to the limit superior and limit inferior of $B$ respectively. Similarly, for topological space $X$ , take the sequence $(x n)$ where $x_n \in X$ for any $n \in \mathbb{N}$ with $\mathbb{N}$ being the set of natural numbers. The filter base ("of tails") generated by this sequence is $C$ defined by