Limit superior and limit inferior

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

<math>\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\,\}.</math>

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

<math>\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\,\}.</math>

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

<math>\liminf_{n\rightarrow\infty}x_n\leq\limsup_{n\rightarrow\infty}x_n.</math>

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

lim inf (p_n+1 - p_n),

where p_n represents 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.

See Borel-Cantelli Lemma for an example.