Lim sup

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

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.

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. However, we often would like to discuss the limit of a sequence without having to spend much time on investigating convergence, or thinking about which subsequence to extract. Therefore, we need to broaden our concept of limits to allow for the possibility of divergent sequences. Definition 3. Then c is called the limit inferior of the sequence. Let be a sequence of real numbers. Then c is called the limit superior of the sequence. If you try to guess the answer quickly, you might get confused between an ordinary supremum and the lim sup , or the regular infimum and the lim inf. Examples 3. What is inf , sup , lim inf and lim sup for? What is inf , sup , lim inf and lim sup for While these limits are often somewhat counter-intuitive, they have one very useful property: Proposition 3. Proof It is important to try to develop a more intuitive understanding about lim sup and lim inf.

The limit superior of the filter base B is defined as. If you try to guess the answer quickly, you might get confused between an ordinary supremum and the lim supor the regular infimum and the lim inf, lim sup. Such set limits are lim sup in measure theory and probability.

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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. On the other hand, there are more general topological notions of set convergence that do involve accumulation points under different metrics or topologies. The two equivalent definitions are as follows. To see the equivalence of the definitions, consider the limit infimum.

Lim sup

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Bounds of a sequence. Main article: Borel—Cantelli lemma. Proposition 3. The set of all cluster points for that filter base is given by. Special Sequences 4. Principles of Mathematical Analysis. Also note that the limit inferior and the limit superior of a set do not have to be elements of the set. See also: Filters in topology. In particular, Bolzano-Weierstrass' theorem can be useful in case the original sequence was bounded. On the other hand, there are more general topological notions of set convergence that do involve accumulation points under different metrics or topologies. Subsequences 3.

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.

Lim Sup and Lim Inf When dealing with sequences there are two choices: the sequence converges the sequence diverges While we know how to deal with convergent sequences, we don't know much about divergent sequences. ISBN X. Set limits, particularly the limit infimum and the limit supremum, are essential for probability and measure theory. The following makes this precise. Wikimedia Commons. This is the definition used in measure theory and probability. Hidden categories: Articles needing additional references from April All articles needing additional references. 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. Toggle limited content width. Limits, Continuity, and Differentiation 7. Unsourced material may be challenged and removed. Tools Tools.