Limit superior and limit inferior

Consider a sequence ${\displaystyle (x_{n})}$ consisting of real numbers. Assume that the limit superior and limit inferior are real numbers (so, not infinite).

• The limit superior of ${\displaystyle x_{n}}$ is the smallest real number ${\displaystyle b}$ such that, for any positive real number ${\displaystyle \varepsilon }$, there exists a natural number ${\displaystyle N}$ such that ${\displaystyle x_{n}<b+\varepsilon }$ for all ${\displaystyle n>N}$. 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 ${\displaystyle b+\varepsilon }$.
• The limit inferior of ${\displaystyle x_{n}}$ is the largest real number ${\displaystyle b}$ such that, for any positive real number ${\displaystyle \varepsilon }$, there exists a natural number ${\displaystyle N}$ such that ${\displaystyle x_{n}>b-\varepsilon }$ for all ${\displaystyle n>N}$. 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 ${\displaystyle b-\varepsilon }$.

In case the sequence is bounded, for all ${\displaystyle \epsilon >0}$ almost all sequence members lie in the open interval ${\displaystyle (\liminf _{n\to \infty }x_{n}-\epsilon ,\limsup _{n\to \infty }x_{n}+\epsilon )}$.

The relationship of limit inferior and limit superior for sequences of real numbers is as follows:

${\displaystyle \limsup _{n\to \infty }(-x_{n})=-\liminf _{n\to \infty }x_{n}}$

As mentioned earlier, it is convenient to extend ${\displaystyle \mathbb {R} }$ to [−∞,∞]. Then, (x_n) in [−∞,∞] converges if and only if

${\displaystyle \liminf _{n\to \infty }x_{n}=\limsup _{n\to \infty }x_{n}}$

in which case ${\displaystyle \lim _{n\to \infty }x_{n}}$ is equal to their common value. (Note that when working just in ${\displaystyle \mathbb {R} }$, convergence to −∞ or ∞ would not be considered as convergence.) Since the limit inferior is at most the limit superior, the following conditions hold

${\displaystyle \liminf _{n\to \infty }x_{n}=\infty \;\;\Rightarrow \;\;\lim _{n\to \infty }x_{n}=\infty ,}$

${\displaystyle \limsup _{n\to \infty }x_{n}=-\infty \;\;\Rightarrow \;\;\lim _{n\to \infty }x_{n}=-\infty .}$

If ${\displaystyle I=\liminf _{n\to \infty }x_{n}}$ and ${\displaystyle S=\limsup _{n\to \infty }x_{n}}$, then the interval [I, S] need not contain any of the numbers x_n, but every slight enlargement [I − ε, S + ε] (for arbitrarily small ε > 0) will contain x_n for all but finitely many indices n. In fact, the interval [I, S] is the smallest closed interval with this property. We can formalize this property like this: there exist subsequences ${\displaystyle x_{k_{n}}}$ and ${\displaystyle x_{h_{n}}}$ of ${\displaystyle x_{n}}$ (where ${\displaystyle k_{n}}$ and ${\displaystyle h_{n}}$ are monotonous) for which we have

${\displaystyle \liminf _{n\to \infty }x_{n}+\epsilon >x_{h_{n}}\;\;\;\;\;\;\;\;\;x_{k_{n}}>\limsup _{n\to \infty }x_{n}-\epsilon }$

On the other hand, there exists a ${\displaystyle n_{0}\in \mathbb {N} }$ so that for all ${\displaystyle n\geq n_{0}}$

${\displaystyle \liminf _{n\to \infty }x_{n}-\epsilon <x_{n}<\limsup _{n\to \infty }x_{n}+\epsilon }$

To recapitulate:

• If ${\displaystyle \Lambda }$ is greater than the limit superior, there are at most finitely many ${\displaystyle x_{n}}$ greater than ${\displaystyle \Lambda }$; if it is less, there are infinitely many.
• If ${\displaystyle \lambda }$ is less than the limit inferior, there are at most finitely many ${\displaystyle x_{n}}$ less than ${\displaystyle \lambda }$; if it is greater, there are infinitely many.

In general we have that

${\displaystyle \inf _{n}x_{n}\leq \liminf _{n\to \infty }x_{n}\leq \limsup _{n\to \infty }x_{n}\leq \sup _{n}x_{n}}$

The liminf and limsup of a sequence are respectively the smallest and greatest cluster points.

• For any two sequences of real numbers ${\displaystyle \{a_{n}\},\{b_{n}\}}$, the limit superior satisfies subadditivity whenever the right side of the inequality is defined (i.e., not ${\displaystyle \infty -\infty }$ or ${\displaystyle -\infty +\infty }$):

${\displaystyle \limsup _{n\to \infty }(a_{n}+b_{n})\leq \limsup _{n\to \infty }(a_{n})+\limsup _{n\to \infty }(b_{n}).}$.

Analogously, the limit inferior satisfies superadditivity:

${\displaystyle \liminf _{n\to \infty }(a_{n}+b_{n})\geq \liminf _{n\to \infty }(a_{n})+\liminf _{n\to \infty }(b_{n}).}$

In the particular case that one of the sequences actually converges, say ${\displaystyle a_{n}\to a}$, then the inequalities above become equalities (with ${\displaystyle \limsup _{n\to \infty }a_{n}}$ or ${\displaystyle \liminf _{n\to \infty }a_{n}}$ being replaced by ${\displaystyle a}$).

• For any two sequences of non-negative real numbers ${\displaystyle \{a_{n}\},\{b_{n}\}}$, the inequalities

${\displaystyle \limsup _{n\to \infty }(a_{n}b_{n})\leq {\biggl (}\limsup _{n\to \infty }a_{n}{\biggr )}{\biggl (}\limsup _{n\to \infty }b_{n}{\biggr )}}$

and

${\displaystyle \liminf _{n\to \infty }(a_{n}b_{n})\geq {\biggl (}\liminf _{n\to \infty }a_{n}{\biggr )}{\biggl (}\liminf _{n\to \infty }b_{n}{\biggr )}}$

hold whenever the right-hand side is not of the form ${\displaystyle 0\cdot \infty }$.

If ${\displaystyle \lim _{n\to \infty }a_{n}=A}$ exists (including the case ${\displaystyle A=+\infty }$) and ${\displaystyle B=\limsup _{n\to \infty }b_{n}}$, then ${\displaystyle \limsup _{n\to \infty }(a_{n}b_{n})=AB}$ provided that ${\displaystyle AB}$ is not of the form ${\displaystyle 0\cdot \infty }$.

• As an example, consider the sequence given by x_n = sin(n). Using the fact that pi is irrational, one can show that

${\displaystyle \liminf _{n\to \infty }x_{n}=-1}$

and

${\displaystyle \limsup _{n\to \infty }x_{n}=+1.}$

(This is because the sequence {1,2,3,...} is equidistributed mod 2π, a consequence of the Equidistribution theorem.)

• An example from number theory is

${\displaystyle \liminf _{n\to \infty }(p_{n+1}-p_{n}),}$

where p_n 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 of April 2014 has only been proven to be less than or equal to 6.[2] The corresponding limit superior is ${\displaystyle +\infty }$, because there are arbitrary gaps between consecutive primes.

In this case, a sequence of sets approaches a limiting set when the elements of each member of the sequence approach the elements of the limiting set. In particular, if {X_n} is a sequence of subsets of X, then:

• lim sup X_n, which is also called the outer limit, consists of those 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 sequence of points x_k and a subsequence {X_nk} of {X_n} such that x_k ∈ X_nk and x_k → x as k → ∞.
• lim inf X_n, which is also called the inner limit, consists of those 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 sequence of points {x_k} such that x_k ∈ X_k and x_k → x as k → ∞.

The limit lim X_n exists if and only if lim inf X_n and lim sup X_n agree, in which case lim X_n = lim sup X_n = lim inf X_n.[4]

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 ∈ X and y ∈ X, the discrete metric is defined by

${\displaystyle d(x,y):={\begin{cases}0&{\text{if }}x=y,\\1&{\text{if }}x\neq y,\end{cases}}}$

under which a sequence of points {x_k} converges to point x ∈ X if and only if x_k = x for all except 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 {X_n} is a sequence of subsets of X, then the following always exist:

• lim sup X_n consists of elements of X which belong to X_n for infinitely many n (see countably infinite). That is, x ∈ lim sup X_n if and only if there exists a subsequence {X_nk} of {X_n} such that x ∈ X_nk for all k.
• lim inf X_n consists of elements of X which belong to X_n for all except finitely many n (i.e., for cofinitely many n). That is, x ∈ lim inf X_n if and only if there exists some m>0 such that x ∈ X_n for all n>m.

Observe that x ∈ lim sup X_n if and only if x ∉ lim inf X_n^c.

• The lim X_n exists if and only if lim inf X_n and lim sup X_n agree, in which case lim X_n = lim sup X_n = lim inf X_n.

In this sense, the sequence has a limit so long as every point in X either appears in all except finitely many X_n or appears in all except finitely many X_n^c. [5]

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, lim inf X_n, is the largest meeting of tails of the sequence, and the outer limit, lim sup X_n, is the smallest joining of tails of the sequence. The following makes this precise.

• Let I_n be the meet of the n^th tail of the sequence. That is,

${\displaystyle {\begin{aligned}I_{n}&=\inf\{X_{m}:m\in \{n,n+1,n+2,\ldots \}\}\\&=\bigcap _{m=n}^{\infty }X_{m}=X_{n}\cap X_{n+1}\cap X_{n+2}\cap \cdots .\end{aligned}}}$

The sequence {I_n} is non-decreasing (I_n ⊆ I_n+1) because each I_n+1 is the intersection of fewer sets than I_n. The least upper bound on this sequence of meets of tails is

${\displaystyle {\begin{aligned}\liminf _{n\to \infty }X_{n}&=\sup\{\inf\{X_{m}:m\in \{n,n+1,\ldots \}\}:n\in \{1,2,\dots \}\}\\&={\bigcup _{n=1}^{\infty }}\left({\bigcap _{m=n}^{\infty }}X_{m}\right).\end{aligned}}}$

So the limit infimum contains all subsets which are lower bounds for all except finitely many sets of the sequence.

• Similarly, let J_n be the join of the n^th tail of the sequence. That is,

${\displaystyle {\begin{aligned}J_{n}&=\sup\{X_{m}:m\in \{n,n+1,n+2,\ldots \}\}\\&=\bigcup _{m=n}^{\infty }X_{m}=X_{n}\cup X_{n+1}\cup X_{n+2}\cup \cdots .\end{aligned}}}$

The sequence {J_n} is non-increasing (J_n ⊇ J_n+1) because each J_n+1 is the union of fewer sets than J_n. The greatest lower bound on this sequence of joins of tails is

${\displaystyle {\begin{aligned}\limsup _{n\to \infty }X_{n}&=\inf\{\sup\{X_{m}:m\in \{n,n+1,\ldots \}\}:n\in \{1,2,\dots \}\}\\&={\bigcap _{n=1}^{\infty }}\left({\bigcup _{m=n}^{\infty }}X_{m}\right).\end{aligned}}}$

So the limit supremum is contained in all subsets which are upper bounds for all except finitely many sets of the sequence.

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:

${\displaystyle \{X_{n}\}=\{\{0\},\{1\},\{0\},\{1\},\{0\},\{1\},\dots \}.}$

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 {X_n} sequence as a whole, and so the interior or inferior limit is the empty set {}. That is,

• lim sup X_n = {0,1}
• lim inf X_n = {}

However, for {Y_n} = {{0},{0},{0},...} and {Z_n} = {{1},{1},{1},...}:

• lim sup Y_n = lim inf Y_n = lim Y_n = {0}
• lim sup Z_n = lim inf Z_n = lim Z_n = {1}

• Consider the set X = {50, 20, -100, -25, 0, 1} and the sequence of subsets:

${\displaystyle \{X_{n}\}=\{\{50\},\{20\},\{-100\},\{-25\},\{0\},\{1\},\{0\},\{1\},\{0\},\{1\},\dots \}.}$

As in the previous two examples,

• lim sup X_n = {0,1}
• lim inf X_n = {}

That is, the four elements that do not match the pattern do not affect the lim inf and lim sup because there are only finitely many of them. In fact, these elements could be placed anywhere in the sequence (e.g., at positions 100, 150, 275, and 55000). 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:

${\displaystyle \{X_{n}\}=\{\{0\},\{1\},\{1/2\},\{1/2\},\{2/3\},\{1/3\},\{3/4\},\{1/4\},\dots \}.}$

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 {X_n} sequence as a whole, and so the interior or inferior limit is the empty set {}. So, as in the previous example,

• lim sup X_n = {0,1}
• lim inf X_n = {}

However, for {Y_n} = {{0},{1/2},{2/3},{3/4},...} and {Z_n} = {{1},{1/2},{1/3},{1/4},...}:

• lim sup Y_n = lim inf Y_n = lim Y_n = {1}
• lim sup Z_n = lim inf Z_n = lim Z_n = {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.[4]^:50–51 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 Ω 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.

The limit inferior of a set X ⊆ Y is the infimum of all of the limit points of the set. That is,

${\displaystyle \liminf X:=\inf\{x\in Y:x{\text{ is a limit point of }}X\}\,}$

Similarly, the limit superior of a set X is the supremum of all of the limit points of the set. That is,

${\displaystyle \limsup X:=\sup\{x\in Y:x{\text{ is a limit point of }}X\}\,}$

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.

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

${\displaystyle \bigcap \{{\overline {B}}_{0}:B_{0}\in B\}}$

where ${\displaystyle {\overline {B}}_{0}}$ is the closure of ${\displaystyle B_{0}}$. 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

${\displaystyle \limsup B:=\sup \bigcap \{{\overline {B}}_{0}:B_{0}\in B\}}$

when that supremum exists. When X has a total order, is a complete lattice and has the order topology,

${\displaystyle \limsup B=\inf\{\sup B_{0}:B_{0}\in B\}.}$

Similarly, the limit inferior of the filter base B is defined as

${\displaystyle \liminf B:=\inf \bigcap \{{\overline {B}}_{0}:B_{0}\in B\}}$

when that infimum exists; if X is totally ordered, is a complete lattice, and has the order topology, then

${\displaystyle \liminf B=\sup\{\inf B_{0}:B_{0}\in B\}.}$

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.

Note that filter bases are generalizations of nets, which are generalizations of sequences. Therefore, these definitions give the limit inferior and limit superior of any net (and thus any sequence) as well. For example, take topological space ${\displaystyle X}$ and the net ${\displaystyle (x_{\alpha })_{\alpha \in A}}$, where ${\displaystyle (A,{\leq })}$ is a directed set and ${\displaystyle x_{\alpha }\in X}$ for all ${\displaystyle \alpha \in A}$. The filter base ("of tails") generated by this net is ${\displaystyle B}$ defined by

${\displaystyle B:=\{\{x_{\alpha }:\alpha _{0}\leq \alpha \}:\alpha _{0}\in A\}.\,}$

Therefore, the limit inferior and limit superior of the net are equal to the limit superior and limit inferior of ${\displaystyle B}$ respectively. Similarly, for topological space ${\displaystyle X}$, take the sequence ${\displaystyle (x_{n})}$ where ${\displaystyle x_{n}\in X}$ for any ${\displaystyle n\in \mathbb {N} }$ with ${\displaystyle \mathbb {N} }$ being the set of natural numbers. The filter base ("of tails") generated by this sequence is ${\displaystyle C}$ defined by

${\displaystyle C:=\{\{x_{n}:n_{0}\leq n\}:n_{0}\in \mathbb {N} \}.\,}$

Therefore, the limit inferior and limit superior of the sequence are equal to the limit superior and limit inferior of ${\displaystyle C}$ respectively.