In
mathematics, the limit inferior and limit 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
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
(see
limit of a function). For a set, they are the
infimum and supremum
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest lo ...
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.
The limit inferior of a sequence
is denoted by
The limit superior of a sequence
is denoted by
Definition for sequences
The of a sequence (''x''
''n'') is defined by
or
Similarly, the of (''x''
''n'') is defined by
or
Alternatively, the notations
and
are sometimes used.
The limits superior and inferior can equivalently be defined using the concept of subsequential limits of the sequence
. An element
of the extended real numbers
is a ''subsequential limit'' of
if there exists a strictly increasing sequence of natural numbers
such that
. If
is the set of all subsequential limits of
, 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
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
. More generally, these definitions make sense in any
partially ordered set
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...
, provided the
suprema and
infima
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest lo ...
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. 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 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
Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions.
These theories are usually studied ...
, 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
In mathematics, the affinely extended real number system is obtained from the real number system \R by adding two infinity elements: +\infty and -\infty, where the infinities are treated as actual numbers. It is useful in describing the algebra o ...
: we add the positive and negative infinities to the real line to give the complete
totally ordered set ��∞,∞ which is a complete lattice.
Interpretation
Consider a sequence
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