Limit superior and limit inferior
In mathematics, the limit inferior (also called infimum limit , liminf , inferior limit , lower limit , or inner limit )and limit superior (also called supremum limit , limsup , superior limit , upper limit , or outer limit ) of a quence can be thought of as limiting (i.e., eventual and extreme) bounds on the quence. The limit inferior and limit superior of a function can be thought of in a similar fashion (e limit of a function). The limit inferior and limit superior of a t are the infimum and supremum of the t's limit points, respectively. In general, when there are multiple objects around which a quence, function, or t 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.
An illustration of limit superior and limit inferior. The quence x n is shown in
blue. The two red curves approach the limit superior and limit inferior of x n , shown
as solid red lines to the right. In this ca, the quence accumulates around the
two limits. The superior limit is the larger of the two, and the inferior limit is the
smaller of the two. The inferior and superior limits only agree when the quence
is convergent (i.e., when there is a single limit).
Definition for quences
The limit inferior of a quence (x n ) is
defined by
or
南美水仙Similarly, the limit superior of (x n
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or
If the terms in the quence are real numbers, the limit superior and limit inferior always exist, as real numbers or ±∞ (i.e., on the extended real number line). More generally, the definitions make n in any partially ordered t,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 cas where the limit does not exist. Whenever lim inf x n and lim sup x n
both exist, we have
Limits inferior/superior are related to big-O notation in that they bound a quence only "in the limit"; the quence may exceed the bound. However, with big-O notation the quence can only exceed the bound in a finite prefix of the quence, whereas the limit superior of a quence like e -n may actually be less than all elements of the quence. The only promi made is that some tail of the quence can be bounded by the limit superior (inferior) plus (minus)
an arbitrarily small positive constant.
The limit superior and limit inferior of a quence are a special ca of tho of a function (e below).
The ca of quences of real numbers
In mathematical analysis, limit superior and limit inferior are important tools for studying quences of real numbers. In order to deal with the difficulties arising from the fact that the supremum and infimum of an unbounded t of real numbers may not exist (the reals are not a complete lattice), it i
置业顾问岗位职责s convenient to consider quences in the affinely extended real number system: we add the positive and negative infinities to the real line to give the complete totally ordered t [-∞,∞], which is a complete lattice.
Interpretation
Consider a quence 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 quence. Only a finite number of elements of the quence are greater than .
•The limit inferior of is the largest real number 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 quence. Only a finite number of elements of the quence are less than .
Properties
The relationship of limit inferior and limit superior for quences of real numbers is as follows
) in [−∞,∞] converges if and only if
As mentioned earlier, it is convenient to extend R to [−∞,∞]. Then, (x
n
in which ca is equal to their common value. (Note that when working just in R, convergence to −∞ or ∞would not be considered as convergence.) Since the limit inferior is at most the limit superior, the condition
implies that
and the condition
implies that
= sin(n). Using the fact that pi is irrational, one can show that
As an example, consider the quence given by x
n
and
(This is becau the quence {1,2,3,...} is equidistributed mod 2π, a conquence of the Equidistribution theorem.) If
and
, but every slight enlargement [I − ε, S + ε] (for then the interval [I, S] need not contain any of the numbers x
n
arbitrarily small ε > 0) will contain x
for all but finitely many indices n. In fact, the interval [I, S] is the smallest
n
clod interval with this property. We can formalize this property like this. If there exists a so that
then there exists a subquence of for which we have that
In the same way, an analogous property holds for the limit inferior: if
then there exists a subquence of for which we have that
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On the other hand we have that if
there exists a so that
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Similarly, if there exists a so that
there exists a so that眼睑炎怎么引起的
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.
In general we have that
The liminf and limsup of a quence are respectively the smallest and greatest cluster points.
An example from number theory is
where p
is the n-th prime number. The value of this limit inferior is conjectured to be 2 - this is the twin prime n
conjecture - but as yet has not even been proved finite. The corresponding limit superior is , becau there are
arbitrary gaps between concutive primes.
•For any two quences of real numbers , the limit superior satisfies superadditivity: (handling appropriately)
Analogously, if is handled with care, the limit inferior satisfies subadditivity
In the particular ca that one of the quences actually converges, say , then the inequalities above
become equalities (with or being replaced by ).
If the limit superior and limit inferior converge to the same value:
Then the limit converges to that value
Real-valued functions
Assume that a function is defined from a subt of the real numbers to the real numbers. As in the ca for quences, 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
红楼梦考点example, given f(x) = sin(1/x), we have lim sup
x→0f(x) = 1 and lim inf
x→0
f(x) = -1. The difference between the two
is a rough measure of how "wildly" the function oscillates, and in obrvation 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 t of measure zero [1]. Note that points of nonzero oscillation (i.e., points at which f is "badly behaved") are discontinuities which, unless they make up a t of zero, are confined to a negligible t.
Functions from metric spaces to metric spaces
There is a notion of lim sup and lim inf for functions defined on a metric space who relationship to limits of real-valued functions mirrors that of the relation between the lim sup, lim inf, and the limit of a real quence. Take metric spaces X and Y, a subspace E contained in X, and a function f : E → Y. The space Y should also be an ordered t, so that the notions of supremum and infimum make n. Define, for any limit point a of E,
and
where B(a;ε) denotes the metric ball of radius ε about a.
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. Take X, Y, E and a as before, but now let X and Y both be topological spaces. In this ca, we replace metric balls with neighborhoods:
(there is a way to write the formula using a lim using nets and the neighborhood filter). This version is often uful in discussions of mi-continuity which crop up in analysis quite often. An interesting note is that this version subsumes the quential version by considering quences as functions from the natural numbers as a topological subspace of the extended real line, into the space (the clo
sure of N in [-∞, ∞] is N∪ {∞}.)
Sequences of ts
The power t ℘(X ) of a t X is a complete lattice that is ordered by t inclusion, and so the supremum and infimum of any t of ts, in terms of t inclusion, of subts always exist. In particular, every subt Y of X is bounded above by X and below by the empty t ∅ becau ∅ ⊆ Y ⊆ X . Hence, it is possible (and sometimes uful) to consider superior and inferior limits of quences in ℘(X ) (i.e., quences of subts of X ).
There are two common ways to define the limit of quences of t. In both cas:
•The quence accumulates around ts of points rather than single points themlves. That is, becau each element of the quence is itlf a t, there exist accumulation ts that are somehow nearby to infinitely many elements of the quence.
•The supremum/superior/outer limit is a t that joins the accumulation ts together. That is, it is the union of all of the accumulation ts. When ordering by t inclusion, the supremum limit is the least upper bound on the t of accumulation points becau it contains each of them. Hence, it is the supremum of the limit points.
•The infimum/inferior/inner limit is a t where all of the accumulation ts meet. That is, it is the interction of all of the accumulation ts. When ordering by t inclusion, the infimum limit is the greatest lower bound on the t of accumulation points becau it is contained in each of them. Hence, it is the infimum of the limit points.•Becau ordering is by t inclusion, then the outer limit will always contain the inner limit (i.e., lim inf X n ⊆lim sup X n ).
The difference between the two definitions involves the topology (i.e., how to quantify paration) is defined. In fact,the cond definition is identical to the first when the discrete metric is ud to induce the topology on X .General t convergence
In this ca, a quence of ts approaches a limiting t when its elements of each member of the quence approach that elements of the limiting t. In particular, if {X n } is a quence of subts of X , then:
•lim sup X n , which is also called the outer limit , consists of tho elements which are limits of points in X n taken from (countably) infinitely many n . That is, x ∈ lim sup X n if and only if there exists a quence of points x k and a subquence {X n k } of {X n } such that x k ∈ X n k and x k → x as k → ∞.
•lim inf X n , which is also called the inner limit , consists of tho elements which are limits of points in X n for all but finitely many n (i.e., cofinitely many n ). That is, x ∈ lim inf X n if and only if there exists a quence of points {x k } such that x k ∈ X k and x k → x as k → ∞.
The limit lim X exists if and only if lim inf X and lim sup X agree, in which ca lim X = lim sup X = lim inf X .[2]Special ca: discrete metric
In this ca, which is frequently ud in measure theory, a quence of ts approaches a limiting t when the limiting t includes elements from each of the members of the quence. That is, this ca specializes the first ca when the topology on t X is induced from the discrete metric. For points x ∈ X and y ∈ X , the discrete metric is defined by
So a quence of points {x k } converges to point x ∈ X if and only if x k = x for all but finitely many k . The following definition is the result of applying this metric to the general definition above.
If {X n } is a quence of subts of X , then:
•lim sup X n consists of elements of X which belong to X n for (countably) infinitely many values of n . That is, x ∈lim sup X n if and only if there exists a subquence {X n k } of {X n } such that x ∈ X n k for all k .
