Limit inferior and limit superior

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.

The limit inferior of a sequence (xn) is defined by

if there exists a strictly increasing sequence of natural numbers

More generally, these definitions make sense in any partially ordered set, provided the suprema and infima exist, such as in a complete lattice.

Whenever lim inf xn and lim sup xn both exist, we have 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.

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.

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.

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

[3] Analogously, the limit inferior satisfies superadditivity:

This idea of oscillation is sufficient to, for example, characterize Riemann-integrable functions as continuous except on a set of measure zero.

[5] Note that points of nonzero oscillation (i.e., points at which f is "badly behaved") are discontinuities which, unless they make up a set of zero, are confined to a negligible set.

There is a notion of limsup and liminf for functions defined on a metric space whose relationship to limits of real-valued functions mirrors that of the relation between the limsup, liminf, and the limit of a real sequence.

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

This finally motivates the definitions for general topological spaces.

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, is N ∪ {∞}.)

There are two common ways to define the limit of sequences of sets.

In both cases: 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.

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.

That is, this case specializes the general definition when the topology on set X is induced from the discrete metric.

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.

In this context, the inner limit, lim inf Xn, is the largest meeting of tails of the sequence, and the outer limit, lim sup Xn, is the smallest joining of tails of the sequence.

They have been broken into sections with respect to the metric used to induce the topology on set X.

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.

The set of all cluster points for that filter base is given by where

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.

The filter base ("of tails") generated by this net is

An illustration of limit superior and limit inferior. The sequence x n is shown in blue. The two red curves approach the limit superior and limit inferior of x n , shown as dashed black lines. In this case, the sequence accumulates around the two limits. The superior limit is the larger of the two, and the inferior limit is the smaller. The inferior and superior limits agree if and only if the sequence is convergent (i.e., when there is a single limit).
In case the sequence is bounded, for all almost all sequence members lie in the open interval