Limit superior and limit inferior
In mathematics Limes superior and limes inferior denote a sequence (xn) the largest or smallest limit of convergent subsequences of (xn ). Analogously Limes superior and limes inferior defined by real-valued functions. Limit superior and limit inferior are a partial substitute for the limit, if it does not exist.
- 5.1 Definition
- 5.2 Properties
- 5.3 Examples
Definition of sequences of real numbers
Formally, the limit inferior is a sequence of real numbers defined as
Or as
Analogously one defines the limit superior of a sequence of real numbers as
For an unlimited downward real sequence of Limes inferior does not exist, analogue does not exist the limit superior for an unlimited upward sequence. For bounded sequences both exist always and then vote with the smallest and largest accumulation point of the sequence match. Limes are often inferior and limit superior, however, considered as elements of the extended real numbers; in this case they always exist.
If there are limes inferior and limit superior of a sequence, then
These definitions are more general sense in a partially ordered set, if the Suprema and infima occurring exist. In a complete lattice these quantities always exist, so that in this case every sequence has a limit inferior and limit superior features.
Sequences of real functions
For a sequence of real functions for the limit inferior and limit superior are defined pointwise, ie
Similarly for lim sup.
One of the most famous mathematical statements that are inferior to use the concept of limit of a sequence of functions, is the lemma of Fatou.
Limit superior and limit inferior of functions
If a real-valued function on a given interval and an interior point of the interval, so Limes superior and limes inferior those values from the extended real numbers, which are defined as follows:
Denotes the image set of the open interval; is to be chosen so small that.
Be a one-sided limit superior and inferior one-sided limit defined analogously to one-sided limits:
Limit superior and limit inferior of functions are used for example in the definition of semicontinuity.
Sequences of sets
Limit superior and limit inferior
For an arbitrary set, the power set forms a complete lattice under the subset relation defined by the order. The limit inferior of a sequence ( An) of arbitrary subsets of is from the set of all elements that are in almost all An. The limit superior of the sequence of sets (An) is from the set of all elements which lie in infinitely many An.
In the language of set theory expressed,
And
The limit superior of sets is used for example in the Borel - Cantelli lemma.
One can clearly make the formulas when you first considered average or union of finite sets. The right side of the equation for the limit superior for and is
In each step, a further quantity divided out from the union of all sets. Back finally remains for all finite n only. In the infinite quantities only remain that occur in infinitely many because they can be divided out never. Thus, the limit superior is just the part that lies in infinitely many.
Associated with sequences of numbers
The characteristic function of the limit inferior and limit superior of sets is the pointwise limit inferior and limit superior of the characteristic functions of the individual quantities: Off
And
Follows
Similarly for lim sup.
Convergence
We say that the sequence (An ) converges to a set A if the limit inferior and the limit superior are the same and writes or. A sequence of subsets of a set converges if and only if there is an index to each, so that valid either for all or for all.
Monotone convergence
Is, then, one can show that (An) converges to the amount and to write.
Accordingly, one can show for that (An) converges to the amount and to write.
Generalization
The definitions of limit superior and limit inferior can be generalized.
Definition
Be an arbitrary topological space, a partially ordered set in which every non-empty subset and there. M carry the induced topology of this order. Let further, and an accumulation point of ( ie every neighborhood of contains an element different from off). The set of environments in will be denoted by.
Define now:
May in this case be replaced by any environment based on.
Properties
- It is always
- Applies
It follows that there exists, and it is
Examples
- For, , and we obtain the well-known from calculus definition of the Limes Limes Superiors inferiors and a sequence of real numbers.
- For, , and we obtain the definition of the Limes Limes Superiors inferiors and for quantity consequences.