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| In [[mathematics]], the '''limit inferior''' and '''limit superior''' of a [[sequence]] can be thought of as limiting (i.e., eventual and extreme) bounds on the sequence. They can be thought of in a similar fashion for a [[function (mathematics)|function]] (see [[limit of a function]]). For a set, they are the [[infimum]] and [[supremum]] of the set's [[limit point]]s, 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.
| | This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users. |
| Limit inferior is also called '''infimum limit''', '''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'''.
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| [[Image:Lim sup example 5.png|right|thumb|300px|An illustration of limit superior and limit inferior. The sequence ''x''<sub>''n''</sub> is shown in blue. The two red curves approach the limit superior and limit inferior of ''x''<sub>''n''</sub>, 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 of the two. The inferior and superior limits agree if and only if the sequence is convergent (i.e., when there is a single limit).]]
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| ==Definition for sequences==
| | Registered users will be able to choose between the following three rendering modes: |
| The limit inferior of a sequence (''x''<sub>''n''</sub>) is defined by
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| :<math>\liminf_{n\to\infty}x_n := \lim_{n\to\infty}\Big(\inf_{m\geq n}x_m\Big)</math> | | '''MathML''' |
| | :<math forcemathmode="mathml">E=mc^2</math> |
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| | <!--'''PNG''' (currently default in production) |
| | :<math forcemathmode="png">E=mc^2</math> |
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| :<math>\liminf_{n\to\infty}x_n := \sup_{n\geq 0}\,\inf_{m\geq n}x_m=\sup\{\,\inf\{\,x_m:m\geq n\,\}:n\geq 0\,\}.</math> | | '''source''' |
| | :<math forcemathmode="source">E=mc^2</math> --> |
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| Similarly, the limit superior of (''x''<sub>''n''</sub>) is defined by
| | <span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples]. |
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| :<math>\limsup_{n\to\infty}x_n := \lim_{n\to\infty}\Big(\sup_{m\geq n}x_m\Big)</math>
| | ==Demos== |
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| | Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]: |
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| :<math>\limsup_{n\to\infty}x_n := \inf_{n\geq 0}\,\sup_{m\geq n}x_m=\inf\{\,\sup\{\,x_m:m\geq n\,\}:n\geq 0\,\}.</math>
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| Alternatively, the notations <math>\varliminf_{n\to\infty}x_n:=\liminf_{n\to\infty}x_n</math> and <math>\varlimsup_{n\to\infty}x_n:=\limsup_{n\to\infty}x_n</math> are sometimes used.
| | * accessibility: |
| | ** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]] |
| | ** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)] |
| | ** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)] |
| | ** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)] |
| | ** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]]. |
| | ** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]]. |
| | ** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode. |
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| If the terms in the sequence are real numbers, the limit superior and limit inferior always exist, as real numbers or ±∞ (i.e., on the [[extended real number line]]). More generally, these definitions make sense in any [[partially ordered set]], provided the [[supremum|suprema]] and [[infimum|infima]] exist, such as in a [[complete lattice]].
| | ==Test pages == |
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| 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. Whenever lim inf ''x''<sub>''n''</sub> and lim sup ''x''<sub>''n''</sub> both exist, we have
| | To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages: |
| | *[[Displaystyle]] |
| | *[[MathAxisAlignment]] |
| | *[[Styling]] |
| | *[[Linebreaking]] |
| | *[[Unique Ids]] |
| | *[[Help:Formula]] |
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| :<math>\liminf_{n\to\infty}x_n\leq\limsup_{n\to\infty}x_n.</math>
| | *[[Inputtypes|Inputtypes (private Wikis only)]] |
| | | *[[Url2Image|Url2Image (private Wikis only)]] |
| Limits inferior/superior are related to [[big-O notation]] in that they bound a sequence only "in the limit"; the sequence may exceed the bound. However, with big-O notation the sequence can only exceed the bound in a finite prefix of the sequence, whereas the limit superior of a sequence like e<sup>−''n''</sup> may actually be less than all elements of the sequence. The only promise made is that some tail of the sequence can be bounded by the limit superior (inferior) plus (minus) an arbitrarily small positive constant.
| | ==Bug reporting== |
| | | If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de . |
| The limit superior and limit inferior of a sequence are a special case of those of a function (see below).
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| == The case of sequences of real numbers ==
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| In [[mathematical analysis]], limit superior and limit inferior are important tools for studying sequences of [[real number]]s. 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.
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| === Interpretation ===
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| Consider a sequence <math>(x_n)</math> consisting of real numbers. Assume that the limit superior and limit inferior are real numbers (so, not infinite).
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| * The limit superior of <math>x_n</math> is the smallest real number <math>b</math> such that, for any positive real number <math>\varepsilon</math>, there exists a [[natural number]] <math>N</math> such that <math>x_n<b+\varepsilon</math> for all <math>n>N</math>. In other words, any number larger than the limit superior is an eventual upper bound for the sequence. Only a finite number of elements of the sequence are greater than <math>b+\varepsilon</math>. | |
| * The limit inferior of <math>x_n</math> is the largest real number <math>b</math> that, for any positive real number <math>\varepsilon</math>, there exists a natural number <math>N</math> such that <math>x_n>b-\varepsilon</math> for all <math>n>N</math>. In other words, any number below the limit inferior is an eventual lower bound for the sequence. Only a finite number of elements of the sequence are less than <math>b-\varepsilon</math>.
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| === Properties ===
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| The relationship of limit inferior and limit superior for sequences of real numbers is as follows
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| :<math>\limsup_{n\to\infty} (-x_n) = -\liminf_{n\to\infty} x_n</math>
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| As mentioned earlier, it is convenient to extend <math>\mathbb{R}</math> to [−∞,∞]. Then, (''x''<sub>''n''</sub>) in [−∞,∞] [[limit of a sequence|converges]] [[if and only if]]
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| :<math>\liminf_{n\to\infty} x_n = \limsup_{n\to\infty} x_n</math>
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| in which case <math>\lim_{n\to\infty} x_n</math> is equal to their common value. (Note that when working just in <math>\mathbb{R}</math>, convergence to −∞ or ∞ would not be considered as convergence.) Since the limit inferior is at most the limit superior, the condition
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| :<math>\liminf_{n\to\infty} x_n = \infty \;\;\Rightarrow\;\; \lim_{n\to\infty} x_n = \infty</math>
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| and the condition
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| :<math>\limsup_{n\to\infty} x_n = - \infty \;\;\Rightarrow\;\; \lim_{n\to\infty} x_n = - \infty.</math>
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| If <math>I = \liminf_{n\to\infty} x_n</math> and <math>S = \limsup_{n\to\infty} x_n</math>, then the interval [''I'', ''S''] need not contain any of the numbers ''x''<sub>''n''</sub>, but every slight enlargement [''I'' − ε, ''S'' + ε] (for arbitrarily small ε > 0) will contain ''x''<sub>''n''</sub> for all but finitely many indices ''n''. In fact, the interval [''I'', ''S''] is the smallest closed interval with this property. We can formalize this property like this: there exist [[subsequence]]s <math>x_{k_n}</math> and <math>x_{h_n}</math> of <math>x_n</math> (where <math>k_n</math> and <math>h_n</math> are monotonous) for which we have
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| :<math>\liminf_{n\to\infty} x_n+\epsilon>x_{h_n} \;\;\;\;\;\;\;\;\; x_{k_n} > \limsup_{n\to\infty} x_n-\epsilon</math>
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| On the other hand, there exists a <math>n_0\in\mathbb{N}</math> so that for all <math>n\geq n_0</math>
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| :<math> \liminf_{n\to\infty} x_n-\epsilon < x_n < \limsup_{n\to\infty} x_n+\epsilon</math>
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| To recapitulate:
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| * If <math>\Lambda</math> is greater than the limit superior, there are at most finitely many <math>x_n</math> greater than <math>\Lambda</math>; if it is less, there are infinitely many. | |
| * If <math>\lambda</math> is less than the limit inferior, there are at most finitely many <math>x_n</math> less than <math>\lambda</math>; if it is greater, there are infinitely many.
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| In general we have that
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| :<math>\inf_n x_n \leq \liminf_{n \to \infty} x_n \leq \limsup_{n \to \infty} x_n \leq \sup_n x_n</math>
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| The liminf and limsup of a sequence are respectively the smallest and greatest [[Limit point|cluster points]].
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| * For any two sequences of real numbers <math>\{a_n\}, \{b_n\}</math>, the limit superior satisfies [[subadditivity]] whenever the right side of the inequality is defined (i.e., not <math>\infty - \infty</math> or <math>-\infty + \infty</math>):
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| :<math>\limsup_{n \to \infty} (a_n + b_n) \leq \limsup_{n \to \infty}(a_n) + \limsup_{n \to \infty}(b_n).</math>.
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| Analogously, the limit inferior satisfies [[superadditivity]]:
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| :<math>\liminf_{n \to \infty} (a_n + b_n) \geq \liminf_{n \to \infty}(a_n) + \liminf_{n \to \infty}(b_n).</math>
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| In the particular case that one of the sequences actually converges, say <math>a_n \to a </math>, then the inequalities above become equalities (with <math>\limsup_{n \to \infty}a_n</math> or <math>\liminf_{n \to \infty}a_n</math> being replaced by <math>a</math>).
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| ==== Examples ====
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| * As an example, consider the sequence given by ''x''<sub>''n''</sub> = [[trigonometric function|sin]](''n''). Using the fact that [[pi]] is [[irrational number|irrational]], one can show that
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| :<math>\liminf_{n\to\infty} x_n = -1</math>
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| :<math>\limsup_{n\to\infty} x_n = +1.</math>
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| (This is because the sequence {1,2,3,...} is [[Equidistributed mod 1|equidistributed mod 2π]], a consequence of the [[Equidistribution theorem]].)
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| * An example from [[number theory]] is
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| :<math>\liminf_{n\to\infty}(p_{n+1}-p_n),</math>
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| where ''p''<sub>''n''</sub> is the ''n''-th [[prime number]].
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| The value of this limit inferior is conjectured to be 2 – this is the [[twin prime conjecture]] – but {{as of|2014|4|lc=y}} has only been proven to be less than or equal to 246.<ref>{{cite web|title=Bounded gaps between primes|url=http://michaelnielsen.org/polymath1/index.php?title=Bounded_gaps_between_primes|website=Polymath wiki|accessdate=14 May 2014}}</ref> The corresponding limit superior is <math>+\infty</math>, because there are arbitrary [[Gaps between prime numbers|gaps between consecutive primes]].
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| == Real-valued functions ==
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| Assume that a function is defined from a subset of the real numbers to the real numbers. As in the case for sequences, the limit inferior and limit superior are always well-defined if we allow the values +∞ and −∞; in fact, if both agree then the limit exists and is equal to their common value (again possibly including the infinities). For example, given ''f''(''x'') = sin(1/''x''), we have lim sup<sub>''x''→''0''</sub> ''f''(''x'') = 1 and lim inf<sub>''x''→''0''</sub> ''f''(''x'') = −1. The difference between the two is a rough measure of how "wildly" the function oscillates, and in observation of this fact, it is called the [[Oscillation (mathematics)|oscillation]] of ''f'' at ''a''. This idea of oscillation is sufficient to, for example, characterize [[Riemann integral|Riemann-integrable]] functions as continuous except on a set of [[measure zero]] [http://tt.lamf.uwindsor.ca/314folder/analbookfiles/RintexistLebesgue.pdf]. Note that points of nonzero oscillation (i.e., points at which ''f'' is "[[pathological (mathematics)|badly behaved]]") are discontinuities which, unless they make up a set of zero, are confined to a negligible set.
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| == Functions from metric spaces to metric spaces == | |
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| There is a notion of lim sup and lim inf for functions defined on a [[metric space]] whose relationship to limits of real-valued functions mirrors that of the relation between the lim sup, lim inf, and the limit of a real sequence. Take metric spaces ''X'' and ''Y'', a subspace ''E'' contained in ''X'', and a function ''f'' : ''E'' → ''Y''. The space ''Y'' should also be an [[Totally ordered set|ordered set]], so that the notions of supremum and infimum make sense. Define, for any [[limit point]] ''a'' of ''E'',
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| :<math>\limsup_{x \to a} f(x) = \lim_{\varepsilon \to 0} ( \sup \{ f(x) : x \in E \cap B(a;\varepsilon)\setminus\{a\} \} ) </math>
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| and
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| :<math>\liminf_{x \to a} f(x) = \lim_{\varepsilon \to 0} ( \inf \{ f(x) : x \in E \cap B(a;\varepsilon)\setminus\{a\} \} ) </math>
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| where ''B''(''a'';ε) denotes the [[Ball (mathematics)|metric ball]] of radius ε about ''a''.
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| Note that as ε shrinks, the supremum of the function over the ball is monotone decreasing, so we have
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| :<math>\limsup_{x\to a} f(x) = \inf_{\varepsilon > 0} (\sup \{ f(x) : x \in E \cap B(a;\varepsilon)\setminus\{a\} \}) </math>
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| and similarly
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| :<math>\liminf_{x\to a} f(x) = \sup_{\varepsilon > 0}(\inf \{ f(x) : x \in E \cap B(a;\varepsilon)\setminus\{a\} \}).</math>
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| This finally motivates the definitions for general topological spaces. Take ''X'', ''Y'', ''E'' and ''a'' as before, but now let ''X'' and ''Y'' both be topological spaces. In this case, we replace metric balls with neighborhoods:
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| :<math>\limsup_{x\to a} f(x) = \inf \{ \sup \{ f(x) : x \in E \cap U\setminus\{a\} \} : U\ \mathrm{open}, a \in U, E \cap U\setminus\{a\} \neq \emptyset \}</math>
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| :<math>\liminf_{x\to a} f(x) = \sup \{ \inf \{ f(x) : x \in E \cap U\setminus\{a\} \} : U\ \mathrm{open}, a \in U, E \cap U\setminus\{a\} \neq \emptyset \}</math>
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| (there is a way to write the formula using a ''lim'' using nets and the neighborhood filter). 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 (−∞,∞) is '''N''' ∪ {∞}.)
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| == Sequences of sets ==
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| The [[power set]] ℘(''X'') of a [[Set (mathematics)|set]] ''X'' is a [[complete lattice]] that is ordered by [[inclusion (set theory)|set inclusion]], and so the supremum and infimum of any set of subsets (in terms of set inclusion) always exist. In particular, every subset ''Y'' of ''X'' is bounded above by ''X'' and below by the empty set ∅ because ∅ ⊆ ''Y'' ⊆ ''X''. Hence, it is possible (and sometimes useful) to consider superior and inferior limits of sequences in ℘(''X'') (i.e., sequences of subsets of ''X'').
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| There are two common ways to define the limit of sequences of sets. In both cases:
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| * The sequence ''accumulates'' around sets of points rather than single points themselves. That is, because each element of the sequence is itself a set, there exist accumulation ''sets'' that are somehow nearby to infinitely many elements of the sequence.
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| * The supremum/superior/outer limit is a set that [[join (mathematics)|join]]s these accumulation sets together. That is, it is the union of all of the accumulation sets. When ordering by set inclusion, the supremum limit is the least upper bound on the set of accumulation points because it ''contains'' each of them. Hence, it is the supremum of the limit points.
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| * The infimum/inferior/inner limit is a set where all of these accumulation sets [[meet (mathematics)|meet]]. That is, it is the intersection of all of the accumulation sets. When ordering by set inclusion, the infimum limit is the greatest lower bound on the set of accumulation points because it is ''contained in'' each of them. Hence, it is the infimum of the limit points.
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| * Because ordering is by set inclusion, then the outer limit will always contain the inner limit (i.e., lim inf ''X''<sub>''n''</sub> ⊆ lim sup ''X''<sub>''n''</sub>). Hence, when considering the convergence of a sequence of sets, it generally suffices to consider the convergence of the outer limit of that sequence.
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| 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''.
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| ===General set convergence===
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| In this case, a sequence of sets approaches a limiting set when the elements of each member of the sequence approach the elements of the limiting set. In particular, if {''X''<sub>''n''</sub>} is a sequence of subsets of ''X'', then:
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| * lim sup ''X''<sub>''n''</sub>, which is also called the '''outer limit''', consists of those elements which are limits of points in ''X''<sub>''n''</sub> taken from [[countably infinite|(countably) infinite]]ly many ''n''. That is, ''x'' ∈ lim sup ''X''<sub>''n''</sub> if and only if there exists a sequence of points ''x''<sub>''k''</sub> and a ''subsequence'' {''X''<sub>''n''<sub>''k''</sub></sub>} of {''X''<sub>''n''</sub>} such that ''x''<sub>''k''</sub> ∈ ''X''<sub>''n''<sub>''k''</sub></sub> and ''x''<sub>''k''</sub> → ''x'' as ''k'' → ∞.
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| * lim inf ''X''<sub>''n''</sub>, which is also called the '''inner limit''', consists of those elements which are limits of points in ''X''<sub>''n''</sub> for all but finitely many ''n'' (i.e., [[cofinitely]] many ''n''). That is, ''x'' ∈ lim inf ''X''<sub>''n''</sub> if and only if there exists a ''sequence'' of points {''x''<sub>''k''</sub>} such that ''x''<sub>''k''</sub> ∈ ''X''<sub>''k''</sub> and ''x''<sub>''k''</sub> → ''x'' as ''k'' → ∞.
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| The limit lim ''X''<sub>''n''</sub> exists if and only if lim inf ''X''<sub>''n''</sub> and lim sup ''X''<sub>''n''</sub> agree, in which case lim ''X''<sub>''n''</sub> = lim sup ''X''<sub>''n''</sub> = lim inf ''X''<sub>''n''</sub>.<ref name="GSTeel09">{{Cite journal
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| |last1=Goebel
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| |first1=Rafal
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| |last2=Sanfelice
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| |first2=Ricardo G.
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| |last3=Teel
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| |first3=Andrew R.
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| |title=Hybrid dynamical systems
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| |journal=IEEE Control Systems Magazine
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| |year=2009
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| |volume=29
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| |issue=2
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| |pages=28–93
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| |doi=10.1109/MCS.2008.931718}}</ref>
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| ===Special case: discrete metric===
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| In this case, which is frequently used in [[measure theory]], a sequence of sets approaches a limiting set when the limiting set includes elements from each of the members of the sequence. That is, this case specializes the first case when the topology on set ''X'' is induced from the [[discrete metric]]. For points ''x'' ∈ ''X'' and ''y'' ∈ ''X'', the discrete metric is defined by
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| :<math>d(x,y) := \begin{cases} 0 &\text{if } x = y,\\ 1 &\text{if } x \neq y. \end{cases}</math>
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| So a sequence of points {''x''<sub>''k''</sub>} converges to point ''x'' ∈ ''X'' if and only if ''x''<sub>''k''</sub> = ''x'' for all but finitely many ''k''. The following definition is the result of applying this metric to the general definition above.
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| If {''X''<sub>''n''</sub>} is a sequence of subsets of ''X'', then:
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| * lim sup ''X''<sub>''n''</sub> consists of elements of ''X'' which belong to ''X''<sub>''n''</sub> for '''infinitely many''' ''n'' (see [[countably infinite]]). That is, ''x'' ∈ lim sup ''X''<sub>''n''</sub> if and only if there exists a subsequence {''X''<sub>''n''<sub>''k''</sub></sub>} of {''X''<sub>''n''</sub>} such that ''x'' ∈ ''X''<sub>''n''<sub>''k''</sub></sub> for all ''k''.
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| * lim inf ''X''<sub>''n''</sub> consists of elements of ''X'' which belong to ''X''<sub>''n''</sub> for '''all but finitely many''' ''n'' (i.e., for [[cofinitely]] many ''n''). That is, ''x'' ∈ lim inf ''X''<sub>''n''</sub> if and only if there exists some ''m''>0 such that ''x'' ∈ ''X''<sub>''n''</sub> for all ''n''>''m''.
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| The limit lim ''X'' exists if and only if lim inf ''X'' and lim sup ''X'' agree, in which case lim ''X'' = lim sup ''X'' = lim inf ''X''.<ref name="Halmos50">{{Cite book
| |
| |title=Measure Theory
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| |last=Halmos
| |
| |first=Paul R.
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| |year=1950
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| |location=Princeton, NJ
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| |publisher=D. Van Nostrand Company, Inc.}}</ref> This definition of the inferior and superior limits is relatively strong because it requires that the elements of the extreme limits also be elements of each of the sets of the sequence.
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| Using the standard parlance of set theory, consider the infimum of a sequence of sets. The infimum is a greatest lower ''bound'' or [[meet (mathematics)|meet]] of a set. In the case of a sequence of sets, the sequence constituents meet at a set that is somehow smaller than each constituent set. [[inclusion (set theory)|Set inclusion]] provides an ordering that allows set intersection to generate a greatest lower bound ∩''X''<sub>''n''</sub> of sets in the sequence {''X''<sub>''n''</sub>}. Similarly, the supremum, which is the least upper bound or [[join (mathematics)|join]], of a sequence of sets is the union ∪''X''<sub>''n''</sub> of sets in sequence {''X''<sub>''n''</sub>}.
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| In this context, the inner limit lim inf ''X''<sub>''n''</sub> is the largest meeting of tails of the sequence, and the outer limit lim sup ''X''<sub>''n''</sub> is the smallest joining of tails of the sequence.
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| | |
| *Let ''I''<sub>''n''</sub> be the meet of the ''n''<sup>th</sup> tail of the sequence. That is,
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| ::<math>I_n := \inf \{ X_m : m \in \{n, n+1, n+2, \ldots\}\} = \bigcap_{m=n}^{\infty} X_m = X_n \cap X_{n+1} \cap X_{n+2} \cap \cdots.</math>
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| :Then ''I''<sub>''k''</sub> ⊆ ''I''<sub>''k''+1</sub> ⊆ ''I''<sub>''k''+2</sub> because ''I''<sub>''k''+1</sub> is the intersection of fewer sets than ''I''<sub>''k''</sub>. In particular, the sequence {''I''<sub>''k''</sub>} is non-decreasing. So the inner/inferior limit is the least upper bound on this sequence of '''meets of tails'''. In particular,
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| ::<math>\begin{align}
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| \liminf_{n\to\infty}X_n &:= \lim_{n\to\infty} \inf\{X_m: m \in \{n, n+1, \ldots\}\}\\
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| &= \sup\{\inf\{X_m: m \in \{n, n+1, \ldots\}\}: n \in \{1,2,\dots\}\}\\
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| &= {\bigcup_{n=1}^\infty}\left({\bigcap_{m=n}^\infty}X_m\right).
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| \end{align}</math>
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| :So the inferior limit acts like a version of the standard infimum that is unaffected by the set of elements that occur only finitely many times. That is, the infimum limit is a subset (i.e., a lower bound) for all but finitely many elements.
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| *Similarly, let ''J''<sub>''m''</sub> be the join of the ''m''<sup>th</sup> tail of the sequence. That is,
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| ::<math>J_m := \sup \{ X_m : m \in \{n, n+1, n+2, \ldots\}\} = \bigcup_{m=n}^{\infty} X_m = X_n \cup X_{n+1} \cup X_{n+2} \cup \cdots.</math>
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| :Then ''J''<sub>''k''</sub> ⊇ ''J''<sub>''k''+1</sub> ⊇ ''J''<sub>''k''+2</sub> because ''J''<sub>''k''+1</sub> is the union of fewer sets than ''J''<sub>''k''</sub>. In particular, the sequence {''J''<sub>''k''</sub>} is non-increasing. So the outer/superior limit is the greatest lower bound on this sequence of '''joins of tails'''. In particular,
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| ::<math>\begin{align}
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| \limsup_{n\to\infty}X_n &:= \lim_{n\to\infty} \sup\{X_m: m \in \{n, n+1, \ldots\}\}\\
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| &= \inf\{\sup\{X_m: m \in \{n, n+1, \ldots\}\}: n \in \{1,2,\dots\}\}\\ | |
| &= {\bigcap_{n=1}^\infty}\left({\bigcup_{m=n}^\infty}X_m\right). | |
| \end{align}</math>
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| :So the superior limit acts like a version of the standard supremum that is unaffected by the set of elements that occur only finitely many times. That is, the supremum limit is a superset (i.e., an upper bound) for all but finitely many elements.
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| The limit lim ''X''<sub>''n''</sub> exists if and only if lim sup ''X''<sub>''n''</sub>=lim inf ''X''<sub>''n''</sub>, and in that case, lim ''X''<sub>''n''</sub>=lim inf ''X''<sub>''n''</sub>=lim sup ''X''<sub>''n''</sub>. In this sense, the sequence has a limit so long as all but finitely many of its elements are equal to the limit.
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| ===Examples===
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| The following are several set convergence examples. They have been broken into sections with respect to the metric used to induce the topology on set ''X''.
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| ; Using the [[discrete metric]]
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| * The [[Borel–Cantelli lemma]] is an example application of these constructs.
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| ; Using either the discrete metric or the [[Euclidean metric]]
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| * Consider the set ''X'' = {0,1} and the sequence of subsets:
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| ::<math>\{X_n\} = \{ \{0\},\{1\},\{0\},\{1\},\{0\},\{1\},\dots \}.</math>
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| :The "odd" and "even" elements of this sequence form two subsequences, <nowiki>{{</nowiki>0},{0},{0},...} and <nowiki>{{</nowiki>1},{1},{1},...}, which have limit points 0 and 1, respectively, and so the outer or superior limit is the set {0,1} of these two points. However, there are no limit points that can be taken from the {''X''<sub>''n''</sub>} sequence as a whole, and so the interior or inferior limit is the empty set {}. That is,
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| :* lim sup ''X''<sub>''n''</sub> = {0,1}
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| :* lim inf ''X''<sub>''n''</sub> = {}
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| :However, for {''Y''<sub>''n''</sub>} = <nowiki>{{</nowiki>0},{0},{0},...} and {''Z''<sub>''n''</sub>} = <nowiki>{{</nowiki>1},{1},{1},...}:
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| :* lim sup ''Y''<sub>''n''</sub> = lim inf ''Y''<sub>''n''</sub> = lim ''Y''<sub>''n''</sub> = {0}
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| :* lim sup ''Z''<sub>''n''</sub> = lim inf ''Z''<sub>''n''</sub> = lim ''Z''<sub>''n''</sub> = {1}
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| * Consider the set ''X'' = {50, 20, -100, -25, 0, 1} and the sequence of subsets:
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| ::<math>\{X_n\} = \{ \{50\}, \{20\}, \{-100\}, \{-25\}, \{0\},\{1\},\{0\},\{1\},\{0\},\{1\},\dots \}.</math>
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| :As in the previous two examples,
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| :* lim sup ''X''<sub>''n''</sub> = {0,1}
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| :* lim inf ''X''<sub>''n''</sub> = {}
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| :That is, the four elements that do not match the pattern do not affect the lim inf and lim sup because there are only finitely many of them. In fact, these elements could be placed anywhere in the sequence (e.g., at positions 100, 150, 275, and 55000). So long as the ''tails'' of the sequence are maintained, the outer and inner limits will be unchanged. The related concepts of ''essential'' inner and outer limits, which use the [[essential supremum]] and [[essential infimum]], provide an important modification that "squashes" countably many (rather than just finitely many) interstitial additions.
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| ; Using the Euclidean metric
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| * Consider the sequence of subsets of [[rational number]]s:
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| ::<math>\{X_n\} = \{ \{0\},\{1\},\{1/2\},\{1/2\},\{2/3\},\{1/3\}, \{3/4\}, \{1/4\}, \dots \}.</math>
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| :The "odd" and "even" elements of this sequence form two subsequences, <nowiki>{{</nowiki>0},{1/2},{2/3},{3/4},...} and <nowiki>{{</nowiki>1},{1/2},{1/3},{1/4},...}, which have limit points 1 and 0, respectively, and so the outer or superior limit is the set {0,1} of these two points. However, there are no limit points that can be taken from the {''X''<sub>''n''</sub>} sequence as a whole, and so the interior or inferior limit is the empty set {}. So, as in the previous example,
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| :* lim sup ''X''<sub>''n''</sub> = {0,1}
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| :* lim inf ''X''<sub>''n''</sub> = {}
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| :However, for {''Y''<sub>''n''</sub>} = <nowiki>{{</nowiki>0},{1/2},{2/3},{3/4},...} and {''Z''<sub>''n''</sub>} = <nowiki>{{</nowiki>1},{1/2},{1/3},{1/4},...}:
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| :* lim sup ''Y''<sub>''n''</sub> = lim inf ''Y''<sub>''n''</sub> = lim ''Y''<sub>''n''</sub> = {1}
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| :* lim sup ''Z''<sub>''n''</sub> = lim inf ''Z''<sub>''n''</sub> = lim ''Z''<sub>''n''</sub> = {0}
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| :In each of these four cases, the elements of the limiting sets are not elements of any of the sets from the original sequence.
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| * The Ω limit (i.e., [[limit set]]) of a solution to a [[dynamic system]] is the outer limit of solution trajectories of the system.<ref name="GSTeel09"/>{{rp|50–51}} Because trajectories become closer and closer to this limit set, the tails of these trajectories ''converge'' to the limit set.
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| :* For example, an LTI system that is the [[cascade connection]] of several [[stability theory|stable]] systems with an undamped second-order [[LTI system]] (i.e., zero [[damping ratio]]) will oscillate endlessly after being perturbed (e.g., an ideal bell after being struck). Hence, if the position and velocity of this system are plotted against each other, trajectories will approach a circle in the [[state space (controls)|state space]]. This circle, which is the Ω limit set of the system, is the outer limit of solution trajectories of the system. The circle represents the locus of a trajectory corresponding to a pure sinusoidal tone output; that is, the system output approaches/approximates a pure tone.
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| ==Generalized definitions==
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| The above definitions are inadequate for many technical applications. In fact, the definitions above are specializations of the following definitions.
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| ===Definition for a set===
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| The limit inferior of a set ''X'' ⊆ ''Y'' is the [[infimum]] of all of the [[limit point]]s of the set. That is,
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| :<math>\liminf X := \inf \{ x \in Y : x \text{ is a limit point of } X \}\,</math>
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| Similarly, the limit superior of a set ''X'' is the [[supremum]] of all of the limit points of the set. That is,
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| :<math>\limsup X := \sup \{ x \in Y : x \text{ is a limit point of } X \}\,</math>
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| 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. In that case every set has a limit superior and a limit inferior. Also note that the limit inferior and the limit superior of a set do not have to be elements of the set.
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| ===Definition for filter bases===
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| Take a [[topological space]] ''X'' and a [[filter base]] ''B'' in that space. The set of all [[cluster point]]s for that filter base is given by
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| :<math>\bigcap \{ \overline{B}_0 : B_0 \in B \}</math>
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| where <math>\overline{B}_0</math> is the [[closure (topology)|closure]] of <math>B_0</math>. This is clearly a [[closed set]] and is similar to the set of limit points of a set. Assume that ''X'' is also a [[partially ordered set]]. The limit superior of the filter base ''B'' is defined as
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| :<math>\limsup B := \sup \bigcap \{ \overline{B}_0 : B_0 \in B \}</math>
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| when that supremum exists. When ''X'' has a [[total order]], is a [[complete lattice]] and has the [[order topology]],
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| :<math>\limsup B = \inf\{ \sup B_0 : B_0 \in B \}</math>
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| Proof:
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| Similarly, the limit inferior of the filter base ''B'' is defined as
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| :<math>\liminf B := \inf \bigcap \{ \overline{B}_0 : B_0 \in B \}</math>
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| when that infimum exists; if ''X'' is totally ordered, is a complete lattice, and has the order topology, then
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| :<math>\liminf B = \sup\{ \inf B_0 : B_0 \in B \}</math>
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| 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.
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| ====Specialization for sequences and nets====
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| Note that filter bases are generalizations of [[net (mathematics)|nets]], which are generalizations of [[sequence]]s. Therefore, these definitions give the limit inferior and [[Net (mathematics)#Limit superior|limit superior]] of any net (and thus any sequence) as well. For example, take topological space <math>X</math> and the net <math>(x_\alpha)_{\alpha \in A}</math>, where <math>(A,{\leq})</math> is a [[directed set]] and <math>x_\alpha \in X</math> for all <math>\alpha \in A</math>. The filter base ("of tails") generated by this net is <math>B</math> defined by
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| :<math>B := \{ \{ x_\alpha : \alpha_0 \leq \alpha \} : \alpha_0 \in A \}.\,</math>
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| Therefore, the limit inferior and limit superior of the net are equal to the limit superior and limit inferior of <math>B</math> respectively. Similarly, for topological space <math>X</math>, take the sequence <math>(x_n)</math> where <math>x_n \in X</math> for any <math>n \in \mathbb{N}</math> with <math>\mathbb{N}</math> being the set of [[natural number]]s. The filter base ("of tails") generated by this sequence is <math>C</math> defined by
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| :<math>C := \{ \{ x_n : n_0 \leq n \} : n_0 \in \mathbb{N} \}.\,</math>
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| Therefore, the limit inferior and limit superior of the sequence are equal to the limit superior and limit inferior of <math>C</math> respectively.
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| ==See also==
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| * [[Essential supremum and essential infimum]]
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| * [[Envelope (waves)]]
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| ==References==
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| {{reflist}}
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| {{refbegin}}
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| *{{cite book
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| | last = Amann
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| | first = H.
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| |author2=Escher, Joachim
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| | title = Analysis
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| | publisher = Basel; Boston: Birkhäuser
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| | year = 2005
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| | pages =
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| | isbn = 0-8176-7153-6
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| }}
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| *{{cite book
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| | last = González
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| | first = Mario O
| |
| | title = Classical complex analysis
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| | publisher = New York: M. Dekker
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| | year = 1991
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| | pages =
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| | isbn = 0-8247-8415-4
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| }}
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| {{refend}}
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| ==External links==
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| * {{springer|title=Upper and lower limits|id=p/u095830}}
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| [[Category:Limits (mathematics)]]
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