Upper and lower limits - Encyclopedia of Mathematics

If {Ak} is a sequence of subsets of X, the upper and lower limit of the sequence {Ak} are defined as lim supk→∞Ak=⋂n∈N⋃k≥nAk,lim infk→∞A ...   Login www.springer.com TheEuropeanMathematicalSociety Navigation Mainpage PagesA-Z StatProbCollection Recentchanges Currentevents Randompage Help Projecttalk Requestaccount Tools Whatlinkshere Relatedchanges Specialpages Printableversion Permanentlink Pageinformation Namespaces Page Discussion Variants Views View Viewsource History Actions Upperandlowerlimits FromEncyclopediaofMathematics Jumpto:navigation, search 2010MathematicsSubjectClassification:Primary:54C05Secondary:54A05[MSN][ZBL] Contents 1Upperandlowerlimitofarealsequence 1.1Definition 1.2Properties 1.3Characterizations 1.4Examples 2Upperandlowerlimitofarealfunction 2.1Definition 2.2Characterizations 2.3Properties 2.4Frommetricspacestosequences 3Upperandlowerlimitofsetsinsettheory 4Lowerlimitofsetsintopology 5References Upperandlowerlimitofarealsequence Definition Theupperandlowerlimitofasequenceofrealnumbers$\{x_n\}$(calledalsolimessuperiorandlimesinferior)canbedefinedinseveralwaysandaredenoted,respectivelyas \[ \limsup_{n\to\infty}\,x_n\qquad\liminf_{n\to\infty}\,\,x_n \] (someauthorsusealsothenotation$\overline{\lim}$and$\underline{\lim}$).Onepossibledefinitionisthefollowing Definition1 \[ \limsup_{n\to\infty}\,x_n=\inf_{n\in\mathbbN}\,\,\sup_{k\geqn}\,x_k \] \[ \liminf_{n\to\infty}\,\,x_n=\sup_{n\in\mathbbN}\,\,\inf_{k\geqn}\,x_k\,. \] Properties Itfollowseasilyfromthedefinitionthat \[ \liminf_n\,\,x_n=-\limsup_n\,(-x_n)\,, \] \[ \liminf_n\,\,(\lambdax_n)=\lambda\,\liminf_n\,\,x_n\qquad\limsup_n\,(\lambdax_n)=\lambda\,\limsup_n\,x_n\qquad\mbox{when}\lambda>0 \] andthat \[ \liminf_n\,\,(x_n+y_n)\geq\liminf\,x_n+\liminf\,\,y_n\qquad\limsup_n\,(x_n+y_n)\leq\limsup\,x_n+\limsup\,y_n \] iftheadditionsarenotofthetype$-\infty+\infty$. Moreover,thelimitof$\{x_n\}$existsanditisarealnumber$L$(respetively$\infty$,$-\infty$)ifandonlyiftheupperandlowerlimitcoincide andarearealnumber$L$(resp.$\infty$,$-\infty$). Theupperandlowerlimitsofasequencearebothfiniteifandonlyifthesequenceisbounded. Characterizations Theupperandlowerlimitscanalsobedefinedinseveralalternativeways.Inparticular Theorem1 Let$S:=\{a\in]-\infty,\infty]:\{k:x_k>a\}\mbox{isfinite}\}$and$L:=\{a\in[-\infty,\infty[:\{k:x_kU$thereis$N\in\mathbbN$suchthat$x_nN$; if$U>-\infty$forall$uN$with$x_k>u$. $L:=\liminfx_n$ischaracterizedbythetwoproperties: if$U>-\infty$forall$uu$forall$n>N$; if$UU$and$N\in\mathbbN$thereisa$k>N$with$x_k0}\,\sup\,\{f(x):|x-x_0|0}\,\inf\,\{f(x):|x-x_0|0}\,\sup\,\{f(x):d(x,x_0)0}\,\inf\,\{f(x):d(x,x_0)