More on the limit inferior, limit superior and now the. Here is a sort of practical way of determining the lim sup of the sequence. Math 316, intro to analysis limits superior and inferior. The concept of inferior limit and superior limit of sequences of real numbers is defined here. We proved abstractly that e 0 must be a closed subset of r. Limit superior and limit inferior 1 limit superior and limit inferior let xn n. Rather than compute the limit inferior and limit superior we could compute the limit, and conclude the latter are the same. A completely identical argument just replace the infs with sups above shows that the limit inferior exists as well. Limit superior and limit inferior throughout this note, x n n2n will be a bounded sequence in r. Ive found that some students have difficulty understanding the usual definitions of limit superior and inferior because these definitions combine the notions of limits, of suprema, and of infima, all of which the student may have learned only recently and not fully internalized. We define the limit superior and limit inferior of the sequence to be. Recall that we defined the limit superior and limit inferior in class as follows.
In this paper we extend concepts ofstatistical limit superior and inferior to i. Our results are analogous to the results of fridy and orhan proc. In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting i. In this paper we study the concept of statistical limit superior and statistical limit inferior in probabilistic normed spaces. Solutions to tutorial 3 week 4 material covered outcomes. Chidume chapter 5 limit inferior and limit superior we. More on the limit inferior, limit superior and now the limit of a function james k. Indeed, it is an essential feature of the theory that we do not impose any requirements implying. As usual in such theories, if the limit superior and limit inferior agree we can write the common value as b lim l x h i, x since we assume so little in general, there is no assumption that the limit superior must exceed the limit inferior. The limit superior and limit inferior a number a is called a limit.
Limit points, limit inferior, and limit superior of sequences winfried just, ohio university this module is based on section 6. Suppose s2 is convergent and s3 is convergent and for every element nof nholds s2n s3n. Pdf inferior limit, superior limit and convergence of. N be a sequence in r and let e0 be its set of subsequential limits in r. Limit superior and limit inferior limit superior, limit. The cluster setlimit superior and inferiorsome important theorems problem show that 12 6 limsupcosn 6 1. Basic properties of limsup and liminf 1 equivalent definitions. The limit inferior and limit superior of a bounded sequence of fuzzy real numbers have been introduced by aytar et al. Ra limit superior, limit inferior, and bolzanoweierstrass. Introductory analysis 2spring 2010 superior and inferior.
The limit inferior and limit superior of a sequence. Introduction the idea of convergence of real sequences had. On ilimit superior and ilimit inferior of sequences in. Pdf inferior limit and superior limit of sequences of. Informally, for a sequence in r, the limit superior, or limsup, of a sequence is the largest subsequential limit. The cluster setlimit superior and inferiorsome important theorems note observethatliminfa n 6 limsupa n. In this section we study bounded sequences and their subsequences. An illustration of limit superior and limit inferior. Compute the limit inferior and limit superior of the following by using the fact that they are the smallest and largest point of accumulation. Pdf ilimit superior and limit inferior researchgate.
Analysis one limit inferrior and limit superior examples example fx sin1x. Informally, for a sequence in r, the limit superior, or lim sup, of a sequence is the largest subsequential limit. The superior limit is the larger of the two, and the inferior limit is the smaller of the two. Similar deduction can be made of the limit inferior of the function. On i limit superior and i limit inferior of sequences in intuitionistic fuzzy normed spaces mausumi sen nit silchar silchar 788010, india abstract in this article we introduce the notions of i limit superior and i limit inferior for sequences in intuitionistic fuzzy normed linear spaces and prove intuitionistic fuzzy analogue of some. Limit superior and inferior the real numbers have two. In math202, we study the limit of some sequences, we also see some theorems related to limit. First we form the set s consisting of all limits of convergent. Since fa kg1 k1 and fb kg 1 k1 are monotone sequences, their. Limit superior and limit inferior wikimedia commons.
Inferior limit, superior limit and convergence of sequences of extended real numbers article pdf available in formalized mathematics 154. Math 3283w worksheet tuesday, april 3 2018 the limit inferior. Pdf statistical limit superior and limit inferior in. However, limit superior and limit inferior can attain infinite values so their codomain should be a set bigger than r.
Limit superior and limit inferior 1 limit superior and limit inferior let xn nn be a sequence in r and let e0 be its. In this paper we give a simplified expressions for limit inferior and limit superior. Limit superior and limit inferior suppose that fxng is a bounded real sequence. Mtht 430 chap 8h limsup and liminf for functions see also spivak chapter 8 problem 18 let x k be a bounded sequence. In this paper we have extended the concepts of i limit superior and limit inferior to statistical limit superior and istatistical limit inferior and studied some of their properties for sequence of real numbers. It turns out that up to a denumerable set, the limit superior is always less than or equal to the right limit superior. We can study the latter by introducing another concept of limits of sequences. Math 1d, week 2 cauchy sequences, limits superior and inferior, and series3 so the limit superior exists. The limit inferior and limit superior of a sequence or a function are specializations of this definition. Arbitrary function, limit superior, dini derivative. Proposition let fa ngbe a sequence and set a fsubsequential limits of a ng.
Media in category limit superior and limit inferior the following 8 files are in this category, out of 8 total. Math301 real analysis 2008 fall limit superior and limit. The limit inferior and limit superior are defined for bounded sequences in two steps. The two red curves approach the limit superior and limit inferior of x n, shown as dashed black lines. In this case, the sequence accumulates around the two limits. A bounded sequence has at least one limit point according to bolzano weierstrass. A properly divergent sequence does not have any limit point. Introduction in order to make us understand the information more on approaches of a given real sequence an n 1, we give two definitions, thier names are upper limit and lower limit. Limit inferior and limit superior are more general terms that represent the infimum and supremum respectively of all limit points of a set. In the following, we will consider extended real number system.
A point x 2 ris called a cluster point of xn if for all 0, there are in. N be a sequence in rand let e 0 be its set of subsequential limits in r. The limit inferior or lim inf of fa ng isgivenby liminfa n lim n. In particular, we define the socalled limit superior and limit inferior of a bounded sequence and talk about limits. A bounded sequence has at least one limit point according to bolzanoweierstrass. Some properties of limit inferior and limit superior for. Research open access statistical limit superior and limit. In this paper we extend the concepts of statistical limit superior and inferior as introduced by fvidy and orhan to astatistical limit superior and inferior and give some astatistical analogue. Precise definition of a limit example 1 linear function duration. Introductory analysis 2spring 2010 superior and inferior limits suppose we have a sequence fang of real numbers. R has a metric which enables us to talk about convergence of sequences and continuity of functions r has an order which enables us.
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