A sequence a n1 n1 converges to 7 if for every 0 there exists n2n such that for every nnwe have ja. The goal is to give a self contained exposition of the notion of completeness of the real number system. Introductory analysis 2spring 2010 superior and inferior limits. In my text book the definition of these two is this. The lim x n exists if and only if lim inf x n and lim sup x n agree, in which case lim x n lim sup x n lim inf x n. The sequence a n is bounded and increasing, so it has a limit. Solutions to tutorial 3 week 4 material covered outcomes. Anyone can explain to me whats the difference between. We define the limit superior and limit inferior of the sequence to be.
Liminf and limsup for bounded sequences of real numbers. Media in category limit superior and limit inferior the following 8 files are in this category, out of 8 total. Although i could use this as the definition of limsup. Note that we have proven that an if lim sup an the corresponding statement for lim inf is that. I think its safe to say what i do here is more of the traditional approach. When trying to find lim sup and lim inf for a given sequence, it is best to find the first few a j s or b j s, respectively, and then to determine the limit of those. Here is a sort of practical way of determining the lim sup of the sequence. Likewise there is a largest number smaller than all numbers in a called infa the in. An oscillation counterpart of this result, as well as the criteria in. Can someone clearly explain about the lim sup and lim inf. Introductory analysis 2spring 2010 superior and inferior. Lim inf and lim sup 3 we have shown that limsupx n is the largest limit of convergent subsequences of x n. Since fa kg1 k1 and fb kg 1 k1 are monotone sequences, their.
Supplement on lim sup and lim inf 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. The lim inf and lim sup and cauchy sequences uc davis. Introductory analysis 2spring 2010 superior and inferior limits suppose we have a sequence fang of real numbers. Whenever lim inf xn and lim sup xn both exist, we have. Limit superior and limit inferior wikimedia commons. Furthermore, 1 is the smallest number which is greater than all of fs values. In general, we may meet some sequences which does not. A sequence a n1 n1 converges to 7 if for every 0 there exists n2n such that for every nnwe have ja n 7j 3333. We define the liminf and limsup of a sequence of events ai i1. Scalar multiplication by a positive constant multiplies the inf or sup, while multiplication by a negative constant switches the inf and sup, proposition 2. Supplement on lim sup and lim inf 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. View limsupliminf from chinese 340 at new york university. Anyone can explain to me whats the difference between limit. 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.
One can easily show optional homework that sup ke n k sup e n 8n 2n. How can we explain limit supremum and limit infimum to students. Ocean acidification impacts oysters memory of environmental stress. A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext. An oscillation counterpart of this result, as well as the criteria in case. Furthermore, 1 is the smallest number which is greater than all of fs.
In the following, we will consider extended real number system. We begin by stating explicitly some immediate properties of the sup and inf, which we use below. Thus f kg k 1 converges towards the in mum of its range. Solution 2 in this assignment, fx ngand fy ngare sequences of real numbers. To understand why the theorem is likely to be correct not proving it, of course, one can just try constructing a counterexample, a. 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. Mtht 430 chap 8h limsup and liminf for functions see also spivak chapter 8 problem 18 let x k be a bounded sequence. If the answer is yes, let n1 be the rst index at which that largest term appears it could appear. We say that eis bounded above if there exists a real number u.
Examples of lim sup and lim inf example calculate lim sup an and lim inf an for an. In math202, we study the limit of some sequences, we also see some theorems related to limit. Informally, for a sequence in r, the limit superior, or limsup, of a sequence is the largest subsequential limit. Notes on sups, infs and sequences pdf ttu math dept texas. 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 particular the results obtained in this section will be essential for establishing properties of distribution functions, the weak law of large numbers. Real numbers, sequences, limits, lim sup, set notation.
The liminf is defined identically, except with inf instead of sup. Anyone can explain to me whats the difference between limit, limsup and liminf of a function. This completes that proof that limsup an is a cluster point. Lim sup and lim inf often play an important investigative role in the proofs of many results in analysis. In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting. The compactness conditions for d 2 are considerably more cumbersome, see 4, 93. Basic properties of limsup and liminf 1 equivalent definitions. Computer model solves mystery of how gas bubbles build big methane hydrate deposits. All about lim sup and lim inf by ng tze beng the use of sequences in analysis is very important in understanding the concepts of continuity, differentiability, compactness and connectedness. Max, min, sup, inf we would like to begin by asking for the maximum of the function fx sinxx. Convergence in this section we discuss several topics that are a bit advanced, but very important. It would be helpful to explain with concrete example.