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In mathematics, the limit inferior and limit superior (also called infimum limit and supremum limit) of a sequence can be thought of as limiting bounds on the sequence. (See limit of a function.) Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ...
This is a page about mathematics. ...
In mathematics, the limit of a function is a fundamental concept in mathematical analysis. ...
The limit inferior of a sequence (xn) is defined as
 or  Similarly, the limit superior of (xn) is defined as  or  These definitions make sense in any partially ordered set, provided the suprema and infima exist. In a complete lattice, the suprema and infima always exist, and so in this case every sequence has a limit superior and a limit inferior. In mathematics, especially order theory, a partially ordered set (or poset for short) is a set equipped with a partial order relation. ...
In mathematics, the supremum of an ordered set S is the least element that is greater than or equal to each element of S. Consequently, it is also referred to as the least upper bound (also lub and LUB). ...
In mathematics the infimum of a subset of some set is the greatest element, not necessarily in the subset, that is smaller than all other elements of the subset. ...
In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum (join) and an infimum (meet). ...
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 xn and lim sup xn both exist, we have
 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 Big O notation is a mathematical notation used to describe the asymptotic behavior of functions. ...
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 (xn) in [-∞,∞] converges if and only if lim inf xn = lim sup xn, in which case lim xn is equal to their common value. (Note that when working just in R, convergence to -∞ or ∞ would not be considered as convergence.) Calculus is a central branch of mathematics, developed from algebra and geometry. ...
In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ...
Infinity is a term with very distinct, separate meanings which arise in theology, philosophy, mathematics and everyday life. ...
In mathematics, a total order or linear order on a set X is any binary relation on X that is antisymmetric, transitive, and total. ...
In mathematics, the concept of a limit is used to describe the behavior of a function, as its argument gets close to either some point, or infinity; or the behavior of a sequences elements, as their index approaches infinity. ...
As an example, consider the sequence given by xn = sin(n). Using the fact that pi is irrational, one can show that lim inf xn = −1 and lim sup xn = +1. In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ...
Lower-case pi The mathematical constant Ï€ is a real number which may be defined as the ratio of a circles circumference (Greek πεÏιφÎÏεια, periphery) to its diameter in Euclidean geometry, and which is in common use in mathematics, physics, and engineering. ...
In mathematics, an irrational number is any real number that is not a rational number, i. ...
If I = lim inf xn and S = lim sup xn, then the interval [I, S] need not contain any of the numbers xn, but every slight enlargement [I − ε, S + ε] (for arbitrarily small ε > 0) will contain xn 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 This article needs to be cleaned up to conform to a higher standard of quality. ...
 where pn 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. In mathematics, a prime number (or prime) is a natural number greater than one whose only positive divisors are one and itself. ...
The twin prime conjecture is a famous problem in number theory that involves prime numbers. ...
Functions from metric and topological spaces to the real numbers There is a notion of lim sup and lim inf for real-valued 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. Given a metric space X, a subspace E contained in X, and f : E → R we define, for a any point in the closure of E: In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ...
In mathematics, the closure of a set S consists of all points which are intuitively close to S. A point which is in the closure of S is a point of closure of S. The notion of closure is in many ways dual to the notion of interior. ...
 and  where B(a,ε) denotes the metric ball of radius ε about a. As in the case for sequences, these are always well-defined if we allow the values +∞ and -∞, and if both are equal 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 supx→0 f(x) = 1 and lim infx→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: points of nonzero oscillation i.e. points at which f is "badly behaved" are confined to a negligible set. If you are having difficulty understanding this article, you might wish to learn more about algebra, functions, and mathematical limits. ...
Let μ be a measure on a sigma algebra Σ of subsets of a set X. An element A in Σ is said to have measure zero if μ(A)=0. ...
In mathematics, a pathological example is one whose properties are (or should be considered) untypically bad. ...
Note that as ε shrinks, the supremum of the function over the ball is monotone decreasing, so we have
 and similarly  This finally motivates the definitions for general topological spaces, for X, E and a as before, but now X only a topological space, we replace balls with neighborhoods:   (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 ∪ {∞}.) In mathematical analysis, semi-continuity (or semicontinuity) is a property of real-valued functions that is weaker than continuity. ...
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 Xn is such a sequence, then an element a of X belongs to lim inf Xn if and only if there exists a natural number n0 such that a is in Xn for all n > n0. The element a belongs to lim sup Xn if and only if for every natural number n0 there exists an index n > n0 such that a is in Xn. In other words, lim sup Xn consists of those elements which are in Xn for infinitely many n, while lim inf Xn consists of those elements which are in Xn for all but finitely many n. In mathematics, given a set S, the power set (or powerset) of S, written or 2S, is the set of all subsets of S. In axiomatic set theory (as developed e. ...
In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ...
In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum (join) and an infimum (meet). ...
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:
 The sequence of n=1,2,3,...,In, where In is the infimum of set n, is non-decreasing, because In⊂In+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:  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.  The limsup is the countable intersection of this non-increasing (each supremum is a subset of the previous supremum) sequence of sets.  See Borel-Cantelli lemma for an example. In probability theory, the Borel_Cantelli lemma is a theorem about sequences of events. ...
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