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Limit superior and limit inferior

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In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting bounds on the sequence.

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

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

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

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

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

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

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

Whenever lim inf x_n and lim sup x_n both exist, then

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

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 lim inf x_n = lim sup x_n, in which case lim x_n is equal to their common value. (Note that when working just in R, convergence to -∞ or ∞ would not be considered as convergence.)

As an example, consider the sequence given by x_n = sin(n). Using the fact that pi is irrational, one can show that lim inf x_n = −1 and lim sup x_n = +1.

If I = lim inf x_n and S = lim sup 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

$\liminf_n(p_{n+1}-p_n),$

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.

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 n=1,2,3,...,I_n, 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 as the opposite. 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.

Categories: Mathematical analysis

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09-23-2007 01:00:40
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