Lim sup

As is the case with sequences of other objects, convergence is not necessary or even usual. More generally, lim sup, 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 lim sup supremum are identical. See below.

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.

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.

If is the sequence of all rational numbers in the interval [0, 1]enumerated in any way, find the lim sup and lim sup inf of that sequence. In fact, lim sup, the second definition is identical to the first when the discrete metric is used to induce the topology on X. In general, when there are multiple objects around which a sequence, lim sup, 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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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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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. The following makes this precise. Sequences of Functions 9. When X has a total order , is a complete lattice and has the order topology ,. The next results will attempt to make these concepts somewhat more clear. Set limits, particularly the limit infimum and the limit supremum, are essential for probability and measure theory. As is the case with sequences of other objects, convergence is not necessary or even usual. See the examples below. 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. In that case every set has a limit superior and a limit inferior. Amann, H. Wikimedia Commons. The Cantor set is defined this way.

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

Similarly, the limit superior of X is the supremum of all of the limit points of the set. This is the definition used in measure theory and probability. Hidden categories: CS1 maint: location missing publisher All articles lacking reliable references Articles lacking reliable references from October Articles with short description Short description matches Wikidata Articles lacking in-text citations from February All articles lacking in-text citations Articles containing potentially dated statements from April All articles containing potentially dated statements Commons category link is on Wikidata. Lim Sup and Lim Inf 3. The final statement relates lim sup and lim inf with our usual concept of limit. More generally, these definitions make sense in any partially ordered set , provided the suprema and infima exist, such as in a complete lattice. February Learn how and when to remove this template message. 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. Main article: Borel—Cantelli lemma. Infinity and Induction 3. The limit superior and limit inferior of a sequence are a special case of those of a function see below. The next results will attempt to make these concepts somewhat more clear.