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The limit inferior (or lower limit) of a sequence (x_{n}) is defined as
Similarly, the limit superior (or upper limit) of (x_{n}) is defined as
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
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 [Iε, 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
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 n_{0} such that a is in X_{n} for all n > n_{0}. The element a belongs to lim sup X_{n} if and only if for every natural number n_{0} there exists an index n > n_{0} 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 BorelCantelli Lemma for an example.
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