Lim sup

One possibility is to try and extract a convergent subsequence, as described in the last section. In particular, Bolzano-Weierstrass' theorem can be useful in case the original sequence was bounded, lim sup. However, we often would like to discuss the limit of a sequence without having to spend much time on investigating convergence, lim sup thinking about which subsequence to extract. Therefore, we need to broaden our concept of limits to allow for the possibility of divergent sequences.

As is the case with sequences of other objects, convergence is not necessary or even usual. More generally, again analogous to real-valued sequences, the less restrictive limit infimum and limit supremum of a set sequence always exist and can be used to determine convergence: the limit exists if the limit infimum and limit supremum are identical. See below. Such set limits are essential in measure theory and probability. This article is restricted to that situation as it is the only one relevant for measure theory and probability. See the examples below.

Lim sup

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 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. 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. 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. The limit superior and limit inferior of a sequence are a special case of those of a function see below. In mathematical analysis , limit superior and limit inferior are important tools for studying sequences of real numbers. Assume that the limit superior and limit inferior are real numbers so, not infinite. Assume that a function is defined from a subset of the real numbers to the real numbers.

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Lim sup

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 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.

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Therefore A j tends to the smallest possible limit of any convergent subsequence. New York: McGraw-Hill. In other projects. ISBN Springer Texts in Statistics. They can be thought of in a similar fashion for a function see limit of a function. Or more formally:. Also note that the limit inferior and the limit superior of a set do not have to be elements of the set. 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. Definition 3. The set of all cluster points for that filter base is given by. On the other hand, there are more general topological notions of set convergence that do involve accumulation points under different metrics or topologies.

As is the case with sequences of other objects, convergence is not necessary or even usual.

Or more formally:. In other projects. Principles of Mathematical Analysis. Proposition 3. Fundamentals of abstract analysis. If is the sequence of all rational numbers in the interval [0, 1] , enumerated in any way, find the lim sup and lim inf of that sequence. Next Previous Glossary Map. Examples 3. One possibility is to try and extract a convergent subsequence, as described in the last section. Assume that X is also a partially ordered set. Boca Raton, FL. The Integral 8. Then c is called the limit superior of the sequence.