In mathematics, the limit inferior and limit superior (or limes inferior and limes superior) of a sequence can be thought of as limiting (that is, eventual and extreme) bounds on the sequence. They can be thought of in a similar fashion for a function (see limit of a function). For a set, they are the infimum and supremum of the set's limit points, respectively. In general, when there are multiple objects around which a sequence, function, or set accumulates, the inferior and superior limits extract the smallest and largest of them; the type of object and the measure of size is context-dependent, but the notion of extreme limits is invariant. Limit inferior is also called infimum limit, limit infimum, liminf, inferior limit, lower limit, or inner limit; limit superior is also known as supremum limit, limit supremum, limsup, superior limit, upper limit, or outer limit.
The limit inferior of a sequence is denoted by
and the limit superior of a sequence is denoted by
Definition for sequences The of a sequence is defined by
or
Similarly, the of is defined by
or
Alternatively, the notations and are sometimes used.
The limits superior and inferior can equivalently be defined using the concept of subsequential limits of the sequence . An element of the extended real numbers is a subsequential limit of if there exists a strictly increasing sequence of natural numbers such that . If is the set of all subsequential limits of , then
and
If the terms in the sequence are real numbers, the limit superior and limit inferior always exist, as the real numbers together with ±∞ (i.e. the extended real number line) are complete. 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. In general,
The limits inferior and 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 above by the limit superior plus an arbitrarily small positive constant, and bounded below by the limit inferior 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).
The case of sequences of real numbers
In mathematical analysis, limit superior and limit inferior are important tools for studying sequences of real numbers. Since the supremum and infimum of an unbounded 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.
Interpretation
Consider a sequence consisting of real numbers. Assume that the limit superior and limit inferior are real numbers (so, not infinite). The limit superior of is the smallest real number such that, for any positive real number , there exists a natural number such that for all . 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 . The limit inferior of is the largest real number such that, for any positive real number , there exists a natural number such that for all . 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 .
Properties
The relationship of limit inferior and limit superior for sequences of real numbers is as follows:
As mentioned earlier, it is convenient to extend to Then, in converges if and only if
in which case is equal to their common value. (Note that when working just in convergence to or would not be considered as convergence.) Since the limit inferior is at most the limit superior, the following conditions hold
If and , then the interval need not contain any of the numbers but every slight enlargement for arbitrarily small will contain for all but finitely many indices In fact, the interval is the smallest closed interval with this property. We can formalize this property like this: there exist subsequences and of (where and are increasing) for which we have
On the other hand, there exists a so that for all
To recapitulate:
If is greater than the limit superior, there are at most finitely many greater than if it is less, there are infinitely many. If is less than the limit inferior, there are at most finitely many less than if it is greater, there are infinitely many.
Conversely, it can also be shown that:
If there are infinitely many greater than or equal to , then is lesser than or equal to the limit supremum; if there are only finitely many greater than , then is greater than or equal to the limit supremum. If there are infinitely many lesser than or equal to , then is greater than or equal to the limit inferior; if there are only finitely many lesser than , then is lesser than or equal to the limit inferior.
In general,The liminf and limsup of a sequence are respectively the smallest and greatest cluster points. For any two sequences of real numbers the limit superior satisfies subadditivity whenever the right side of the inequality is defined (that is, not or ):
Analogously, the limit inferior satisfies superadditivity:In the particular case that one of the sequences actually converges, say then the inequalities above become equalities (with or being replaced by ).
For any two sequences of non-negative real numbers the inequalities and hold whenever the right-hand side is not of the form
If exists (including the case ) and then provided that is not of the form
Examples
As an example, consider the sequence given by the sine function: Using the fact that π is irrational, it follows that and (This is because the sequence is equidistributed mod 2π, a consequence of the equidistribution theorem.)
An example from number theory is where is the -th prime number. The value of this limit inferior is conjectured to be 2 – this is the twin prime conjecture – but has only been proven to be less than or equal to 246. The corresponding limit superior is , because there are arbitrarily large gaps between consecutive primes.
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 , we have and . 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 0. This idea of oscillation is sufficient to, for example, characterize Riemann-integrable functions as continuous except on a set of measure zero. Note that points of nonzero oscillation (i.e., points at which f is "badly behaved") are discontinuities which make up a negligible subset.
The limit superior of a real-valued function defined on an interval containing a point is
and the limit inferior is
Moreover, there are one-sided versions for functions which are defined on intervals having as an endpoint:
Functions from topological spaces to complete lattices
Functions from metric spaces
There is a notion of limsup and liminf for functions defined on a metric space whose relationship to limits of real-valued functions mirrors that of the relation between the limsup, liminf, and the limit of a real sequence. Take a metric space , a subspace contained in , and a function . Define, for any point of the closure of ,
and
where denotes the metric ball of radius about .
Note that as ε shrinks, the supremum of the function over the ball is non-increasing (strictly decreasing or remaining the same), so we have
and similarly
The definition of limsup and liminf in a metric spaces can be equivalently reformulated as follows: The limit superior of as is the supremum of taken over all sequences tending to . The limit inferior of as is the infimum of taken over all sequences tending to .
Functions from topological spaces
This finally motivates the definitions for general topological spaces. Take X, E and a as before, but now let X be a topological space. In this case, we replace metric balls with neighborhoods:
(there is a way to write the formula using "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 [−∞,∞], the extended real number line, is N ∪ {∞}.)
Sequences of sets
The power set ℘(X) of a set X is a complete lattice that is ordered by set inclusion, and so the supremum and infimum of any set of subsets (in terms of set inclusion) always exist. In particular, every subset Y of X is bounded above by X and below by the empty set ∅ because ∅ ⊆ Y ⊆ X. Hence, it is possible (and sometimes useful) to consider superior and inferior limits of sequences in ℘(X) (i.e., sequences of subsets of X).
There are two common ways to define the limit of sequences of sets. In both cases: The sequence accumulates around sets of points rather than single points themselves. That is, because each element of the sequence is itself a set, there exist accumulation sets that are somehow nearby to infinitely many elements of the sequence. The supremum/superior/outer limit is a set that joins these accumulation sets together. That is, it is the union of all of the accumulation sets. When ordering by set inclusion, the supremum limit is the least upper bound on the set of accumulation points because it contains each of them. Hence, it is the supremum of the limit points. The infimum/inferior/inner limit is a set where all of these accumulation sets meet. That is, it is the intersection of all of the accumulation sets. When ordering by set inclusion, the infimum limit is the greatest lower bound on the set of accumulation points because it is contained in each of them. Hence, it is the infimum of the limit points. Because ordering is by set inclusion, then the outer limit will always contain the inner limit (i.e., lim inf Xn ⊆ lim sup Xn). Hence, when considering the convergence of a sequence of sets, it generally suffices to consider the convergence of the outer limit of that sequence. The difference between the two definitions involves how the topology (i.e., how to quantify separation) is defined. In fact, the second definition is identical to the first when the discrete metric is used to induce the topology on X.
General set convergence
A sequence of sets in a metrizable space approaches a limiting set when the elements of each member of the sequence approach the elements of the limiting set. In particular, if is a sequence of subsets of then: which is also called the outer limit, consists of those elements which are limits of points in taken from (countably) infinitely many That is, if and only if there exists a sequence of points and a of such that and