Lim sup
As is the case with sequences of other objects, convergence is not lim sup 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 52x32 determine convergence: the limit exists if the limit infimum and limit supremum are identical. See below, 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.
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.
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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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Toggle limited content width. See the examples below. Such set limits are essential in measure theory and probability. Bounds of a sequence. Retrieved Download as PDF Printable version. The two equivalent definitions are as follows. 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. 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. Take a topological space X and a filter base B in that space. 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.
ISBN X. Special Sequences 4. 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. University of Windsor. Similarly, B j picks the smallest upper bound of the truncated sequences, and hence tends to the greatest possible limit of any convergent subsequence. Set limits, particularly the limit infimum and the limit supremum, are essential for probability and measure theory. The following makes this precise. When X has a total order , is a complete lattice and has the order topology ,. Download as PDF Printable version. Tools Tools. Series of Numbers 5.
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