Wikipedia / Limit inferior and limit superior
Wikipedia / Limit inferior and limit superior
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.
Definition for sequences
The
Similarly, the
are sometimes used.
. of the extended real numbers is a subsequential limit if there exists a strictly increasing sequence of natural numbers, 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
consisting of real numbers. Assume that the limit superior and limit inferior are real numbers (so, not infinite).
.
.
Properties
is the smallest closed interval with this property. We can formalize this property like this: there exist subsequences
To recapitulate:
if it is less, there are infinitely many.
if it is greater, there are infinitely many.
Conversely, it can also be shown that:
is greater than or equal to the limit supremum.
is lesser than or equal to the limit inferior.
The liminf and limsup of a sequence are respectively the smallest and greatest cluster points.
the limit superior satisfies subadditivity
Analogously, the limit inferior satisfies superadditivity).
Examples
As an example, consider the sequence given by the sine Using the fact that π is irrational is equidistributed mod 2π, a consequence of the equidistribution theorem.)
An example from number theory-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., because there are arbitrarily large gaps between consecutive primes.
Real-valued functions
Assume that a function is defined from a subset. 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.
is
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,
and
denotes the metric ball.
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
The definition of limsup and liminf in a metric spaces can be equivalently reformulated as follows:
.
.
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, isN∪{∞}.)
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 joinss 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., liminfXn ⊆ limsupXn). 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 then:
which is also called the outer limit taken from (countably) infinitely and a
which is also called the inner limit (that is, cofinitely if and only if there exists a
The outer and inner limits should not be confused with the set-theoretic limits superior and inferior, as the latter sets are not sensitive to the topological structure of the space.
Special case: discrete metric
This is the definition used in measure theory and probability. Further discussion and examples from the set-theoretic point of view, as opposed to the topological point of view discussed below, are at set-theoretic limit.
By this definition, a sequence of sets approaches a limiting set when the limiting set includes elements which are in all except finitely many sets of the sequence and does not include elements which are in all except finitely many complements of sets of the sequence. That is, this case specializes the general definition when the topology on set X is induced from the discrete metric.
Specifically, for points x, y ∈ X, the discrete metric is defined by
under which a sequence of points (xk) converges to point x ∈ X if and only if xk = x for all but finitely many k. Therefore, if the limit set exists it contains the points and only the points which are in all except finitely many of the sets of the sequence. Since convergence in the discrete metric is the strictest form of convergence (i.e., requires the most), this definition of a limit set is the strictest possible.
If (Xn) is a sequence of subsets of X, then the following always exist:
limsupXn consists of elements of X which belong to Xn for infinitely many n (see countably infinite). That is, x ∈ limsupXn if and only if there exists a subsequence (Xnk) of (Xn) such that x ∈ Xnk for all k.
liminfXn consists of elements of X which belong to Xn for all except finitely many n (i.e., for cofinitely many n). That is, x ∈ liminfXn if and only if there exists some m > 0 such that x ∈ Xn for all n > m.
Observe that x ∈ limsupXn if and only if x ∉ liminfXnc.
limXn exists if and only if liminfXn and limsupXn agree, in which case limXn = limsupXn = liminfXn.
In this sense, the sequence has a limit so long as every point in X either appears in all except finitely many Xn or appears in all except finitely many Xnc.
Using the standard parlance of set theory, set inclusion provides a partial ordering on the collection of all subsets of X that allows set intersection to generate a greatest lower bound and set union to generate a least upper bound. Thus, the infimum or meet of a collection of subsets is the greatest lower bound while the supremum or join is the least upper bound. In this context, the inner limit, liminfXn, is the largest meeting of tails of the sequence, and the outer limit, limsupXn, is the smallest joining of tails of the sequence. The following makes this precise.
Let In be the meet of the nth tail of the sequence. That is,
The sequence (In) is non-decreasing (i.e. In In+1) because each In+1 is the intersection of fewer sets than In. The least upper bound on this sequence of meets of tails is
So the limit infimum contains all subsets which are lower bounds for all but finitely many sets of the sequence.
Similarly, let Jn be the join of the nth tail of the sequence. That is,
The sequence (Jn) is non-increasing (i.e. Jn ⊇ Jn+1) because each Jn+1 is the union of fewer sets than Jn. The greatest lower bound on this sequence of joins of tails is
So the limit supremum is contained in all subsets which are upper bounds for all but finitely many sets of the sequence.
Examples
The following are several set convergence examples. They have been broken into sections with respect to the metric used to induce the topology on set X.
Using the discrete metric
The Borel–Cantelli lemma is an example application of these constructs.
Using either the discrete metric or the Euclidean metric
Consider the set X = {0,1} and the sequence of subsets:
The "odd" and "even" elements of this sequence form two subsequences, ({0}, {0}, {0}, ...) and ({1}, {1}, {1}, ...), which have limit points 0 and 1, respectively, and so the outer or superior limit is the set {0,1} of these two points. However, there are no limit points that can be taken from the (Xn) sequence as a whole, and so the interior or inferior limit is the empty set { }. That is,
limsupXn = {0,1}
liminfXn = { }
However, for (Yn) = ({0}, {0}, {0}, ...) and (Zn) = ({1}, {1}, {1}, ...):
limsupYn = liminfYn = limYn = {0}
limsupZn = liminfZn = limZn = {1}
Consider the set X = {50, 20, −100, −25, 0, 1} and the sequence of subsets:
As in the previous two examples,
limsupXn = {0,1}
liminfXn = { }
That is, the four elements that do not match the pattern do not affect the liminf and limsup because there are only finitely many of them. In fact, these elements could be placed anywhere in the sequence. So long as the tails of the sequence are maintained, the outer and inner limits will be unchanged. The related concepts of essential inner and outer limits, which use the essential supremum and essential infimum, provide an important modification that "squashes" countably many (rather than just finitely many) interstitial additions.
Using the Euclidean metric
Consider the sequence of subsets of rational numbers:
The "odd" and "even" elements of this sequence form two subsequences, ({0}, {1/2}, {2/3}, {3/4}, ...) and ({1}, {1/2}, {1/3}, {1/4}, ...), which have limit points 1 and 0, respectively, and so the outer or superior limit is the set {0,1} of these two points. However, there are no limit points that can be taken from the (Xn) sequence as a whole, and so the interior or inferior limit is the empty set { }. So, as in the previous example,
limsupXn = {0,1}
liminfXn = { }
However, for (Yn) = ({0}, {1/2}, {2/3}, {3/4}, ...) and (Zn) = ({1}, {1/2}, {1/3}, {1/4}, ...):
limsupYn = liminfYn = limYn = {1}
limsupZn = liminfZn = limZn = {0}
In each of these four cases, the elements of the limiting sets are not elements of any of the sets from the original sequence.
The Ω limit (i.e., limit set) of a solution to a dynamic system is the outer limit of solution trajectories of the system. Because trajectories become closer and closer to this limit set, the tails of these trajectories converge to the limit set.
For example, an LTI system that is the cascade connection of several stable systems with an undamped second-order LTI system (i.e., zero damping ratio) will oscillate endlessly after being perturbed (e.g., an ideal bell after being struck). Hence, if the position and velocity of this system are plotted against each other, trajectories will approach a circle in the state space. This circle, which is the limit set of the system, is the outer limit of solution trajectories of the system. The circle represents the locus of a trajectory corresponding to a pure sinusoidal tone output; that is, the system output approaches/approximates a pure tone.
Generalized definitions
The above definitions are inadequate for many technical applications. In fact, the definitions above are specializations of the following definitions.
Definition for a set
The limit inferior of a set X⊆Y is the infimum of all of the limit points of the set. That is,
Similarly, the limit superior of X is the supremum of all of the limit points of the set. That is,
Note that the set X needs to be defined as a subset of a partially ordered set Y that is also a topological space in order for these definitions to make sense. Moreover, it has to be a complete lattice so that the suprema and infima always exist. In that case every set has a limit superior and a limit inferior. Also note that the limit inferior and the limit superior of a set do not have to be elements of the set.
Definition for filter bases
Take a topological space X and a filter base B in that space. The set of all cluster points for that filter base is given by
is the closure. This is clearly a closed set and is similar to the set of limit points of a set. Assume that X is also a partially ordered set. The limit superior of the filter base B is defined as
when that supremum exists. When X has a total order, is a complete lattice and has the order topology,
Similarly, the limit inferior of the filter base B is defined as
when that infimum exists; if X is totally ordered, is a complete lattice, and has the order topology, then
If the limit inferior and limit superior agree, then there must be exactly one cluster point and the limit of the filter base is equal to this unique cluster point.
Specialization for sequences and nets
Note that filter bases are generalizations of nets, which are generalizations of sequences. Therefore, these definitions give the limit inferior and limit superior is a directed set defined by
defined by
respectively.