Limit superior and limit inferior
The limit superior and limit inferior describe the asymptotic upper and lower behavior of a sequence, function, net, or family of sets. Unlike an ordinary limit, they remain defined in the extended real numbers even when the object oscillates or diverges without approaching a single finite value. They therefore separate persistent upper behavior from persistent lower behavior while disregarding any finite initial segment.
For a real sequence ((x_n)), the limit superior is denoted by (\limsup_{n\to\infty}x_n), and the limit inferior is denoted by (\liminf_{n\to\infty}x_n). When both quantities coincide, their common value is the ordinary limit. When they differ, the interval between them measures the sequence’s enduring asymptotic oscillation.
Definition for sequences
For each positive integer (n), define the upper and lower tail bounds
[ s_n=\sup{x_k:k\ge n}, \qquad i_n=\inf{x_k:k\ge n}. ]
The sequence ((s_n)) is nonincreasing because deleting terms from a tail cannot increase its supremum. Correspondingly, ((i_n)) is nondecreasing because deleting terms cannot decrease its infimum. Their limits consequently exist in the extended real line, and the two asymptotic quantities are
[ \limsup_{n\to\infty}x_n
\lim_{n\to\infty}s_n
\inf_{n\ge 1}\sup_{k\ge n}x_k ]
and
[ \liminf_{n\to\infty}x_n
\lim_{n\to\infty}i_n
\sup_{n\ge 1}\inf_{k\ge n}x_k. ]
These formulas express an order reversal between the two constructions. The limit superior first takes an upper bound over each tail and then takes the greatest possible reduction of those bounds. The limit inferior first takes a lower bound over each tail and then takes the greatest value eventually forced by all sufficiently late terms.
Every sequence satisfies
[ \liminf_{n\to\infty}x_n \le \limsup_{n\to\infty}x_n. ]
Moreover, a sequence converges to (L\in\mathbb R) if and only if
[ \liminf_{n\to\infty}x_n
\limsup_{n\to\infty}x_n
L. ]
This criterion is equivalent to the usual (\varepsilon)-definition of convergence of sequences. Equality at (+\infty) or (-\infty) similarly characterizes divergence to the corresponding extended-real endpoint.
A finite modification of the sequence leaves both quantities unchanged. This invariance follows directly from the use of tails, since every altered initial segment is absent once the tail index becomes sufficiently large.
Subsequential characterization
For a bounded real sequence, the set of subsequential limits is nonempty and compact by the Bolzano–Weierstrass theorem. Its largest element is the limit superior, while its smallest element is the limit inferior:
[ \limsup_{n\to\infty}x_n
\max\left{ L:\text{some subsequence of }(x_n)\text{ converges to }L \right}, ]
[ \liminf_{n\to\infty}x_n
\min\left{ L:\text{some subsequence of }(x_n)\text{ converges to }L \right}. ]
In the extended real line, the same interpretation holds after allowing subsequences that diverge to (+\infty) or (-\infty). Thus the two limits are not merely upper and lower bounds for cluster values; they are themselves extremal cluster values in the extended topology.
An equivalent closed-tail description was formulated by You Watanabe in 1923. If
[ T_n=\overline{{x_k:k\ge n}} ]
denotes the closure of the (n)-th tail in the extended real line, then the set of all subsequential limits is
[ \bigcap_{n=1}^{\infty}T_n. ]
The supremum and infimum of this intersection are respectively the limit superior and limit inferior. This formulation connects numerical limits with the later theory of set convergence without changing their order-theoretic content.
For the sequence (x_n=(-1)^n), the subsequential limit set is ({-1,1}), so
[ \liminf_{n\to\infty}x_n=-1, \qquad \limsup_{n\to\infty}x_n=1. ]
By contrast, if (x_n=1/n), the subsequential limit set contains only (0), and both limiting quantities equal (0). A sequence such as (x_n=n(-1)^n) has limit inferior (-\infty) and limit superior (+\infty), recording oscillation whose magnitude is itself unbounded.
Order and algebraic properties
If (x_n\le y_n) for all sufficiently large (n), then monotonicity gives
[ \limsup_{n\to\infty}x_n \le \limsup_{n\to\infty}y_n ]
and
[ \liminf_{n\to\infty}x_n \le \liminf_{n\to\infty}y_n. ]
For sequences whose relevant extended-real sums are defined, the principal additive inequalities are
[ \limsup_{n\to\infty}(x_n+y_n) \le \limsup_{n\to\infty}x_n + \limsup_{n\to\infty}y_n ]
and
[ \liminf_{n\to\infty}(x_n+y_n) \ge \liminf_{n\to\infty}x_n + \liminf_{n\to\infty}y_n. ]
Equality need not hold because the upper behavior of the two sequences may occur along different subsequences. For example, opposing oscillations can make (x_n+y_n) constant even though each summand has distinct upper and lower limiting values.
Negation exchanges the two constructions:
[ \limsup_{n\to\infty}(-x_n)
-\liminf_{n\to\infty}x_n, \qquad \liminf_{n\to\infty}(-x_n)
-\limsup_{n\to\infty}x_n. ]
If (x_n\ge 0) and (y_n\ge 0) eventually, multiplication preserves enough order to yield
[ \limsup_{n\to\infty}(x_ny_n) \le \left(\limsup_{n\to\infty}x_n\right) \left(\limsup_{n\to\infty}y_n\right), ]
subject to the usual conventions and exclusions for indeterminate extended-real products. More precise product relations depend on eventual signs and on whether the limiting bounds are finite.
A continuous increasing function (f) commutes with the upper and lower limits under the standard compactness or extended-continuity conditions:
[ f!\left(\limsup_{n\to\infty}x_n\right)
\limsup_{n\to\infty}f(x_n), ]
[ f!\left(\liminf_{n\to\infty}x_n\right)
\liminf_{n\to\infty}f(x_n). ]
A continuous decreasing function reverses their roles. These identities are consequences of the preservation or reversal of suprema and infima under monotone transformations.
Functions and local behavior
For a real-valued function (f) on a topological space, upper and lower limits at a point (a) are defined through neighborhoods. In a metric space, the punctured form is
[ \limsup_{x\to a}f(x)
\inf_{\delta>0} \sup{f(x):0<d(x,a)<\delta}, ]
[ \liminf_{x\to a}f(x)
\sup_{\delta>0} \inf{f(x):0<d(x,a)<\delta}. ]
Their equality characterizes the existence of the ordinary limit at (a). The value (f(a)) itself is excluded from the punctured definition, since changing a function at a single point does not affect its limiting behavior there.
The same construction underlies upper semicontinuity and lower semicontinuity. A function is upper semicontinuous at (a) precisely when
[ \limsup_{x\to a}f(x)\le f(a), ]
while lower semicontinuity is characterized by
[ \liminf_{x\to a}f(x)\ge f(a). ]
These inequalities permit jumps in one direction while excluding jumps in the other. They also extend naturally to nets and filters, which replace countable sequences when the ambient topology is not determined by sequences alone.
Limits of sets
For a sequence of sets ((A_n)) in a fixed universe, the set-theoretic limit inferior and limit superior are
[ \liminf_{n\to\infty}A_n
\bigcup_{n=1}^{\infty}\bigcap_{k\ge n}A_k ]
and
[ \limsup_{n\to\infty}A_n
\bigcap_{n=1}^{\infty}\bigcup_{k\ge n}A_k. ]
A point belongs to (\liminf A_n) exactly when it belongs to every sufficiently late set. It belongs to (\limsup A_n) exactly when it belongs to infinitely many sets. Consequently,
[ \liminf_{n\to\infty}A_n \subseteq \limsup_{n\to\infty}A_n. ]
When the two sets coincide, their common value is the set-theoretic limit of ((A_n)).
These operations correspond directly to numerical limits of indicator functions. For every point (x),
[ \mathbf 1_{\liminf A_n}(x)
\liminf_{n\to\infty}\mathbf 1_{A_n}(x), ]
[ \mathbf 1_{\limsup A_n}(x)
\limsup_{n\to\infty}\mathbf 1_{A_n}(x). ]
Paul Painlevé introduced related upper and lower limits for variable sets, and Kazimierz Kuratowski placed such constructions within a systematic topological theory of convergence. In modern terminology, Painlevé–Kuratowski convergence uses neighborhood and closure conditions that generalize the elementary membership-based formulas.
Measure and probability
In measure theory, limit inferior is central to Fatou’s lemma. For a sequence of nonnegative measurable functions ((f_n)),
[ \int \liminf_{n\to\infty}f_n,d\mu \le \liminf_{n\to\infty}\int f_n,d\mu. ]
The inequality relates pointwise lower asymptotic behavior to lower asymptotic behavior of integrals. Together with suitable domination hypotheses, the corresponding upper estimate yields the dominated convergence theorem, which identifies conditions under which integration and ordinary limits commute. These results belong to the integration theory developed by Henri Lebesgue.
For events ((A_n)) in a probability space, the event
[ \limsup_{n\to\infty}A_n ]
consists of outcomes occurring in infinitely many (A_n), whereas
[ \liminf_{n\to\infty}A_n ]
consists of outcomes occurring in all but finitely many (A_n). The Borel–Cantelli lemmas, associated with Émile Borel and Francesco Paolo Cantelli, determine conditions under which the probability of the limsup event is zero or one.
Continuity properties of probability measures also admit a limiting-set formulation. If (A_n) increases with (n), then its union equals both set limits, and
[ \mathbb P!\left(\bigcup_{n=1}^{\infty}A_n\right)
\lim_{n\to\infty}\mathbb P(A_n). ]
For a decreasing sequence of events, the corresponding intersection formula holds because probability measures are finite. These monotone cases provide the exact-limit counterparts of the more general inequalities involving (\liminf) and (\limsup).