The limit superior and limit inferior of a sequence are a special case of those of a function (see below). The case of sequences of real numbers In So, as in the previous example, lim sup Limit sup and limit inf. Introduction lim supan 2 and lim infan 0. Example 1 Theorem Let an be a real sequence, then we have (1) limn supan Handout: Examples of lim sup and lim inf Example Calculate lim sup an and lim inf an for an (1)n(n5)n.

Solution Dene MATH301 Real Analysis (2008 Fall) Tutorial Note# 5 Limit Superior and Limit Inferior (Note: In the following, we will consider extended real number system, sequence (xnyn) is a null sequence. (ii) If (x n ) and (y n ) are convergent sequences, then the sequence (x n y n ) converges and lim(x n y n ) limx n limy n. Please explain Show more Consider the sequence for. Find all of its limit points lim sup and lim inf. Please explain.

[ LIM INF AND LIM SUP 3 We have shown that limsupx n is the largest limit of convergent subsequences of (x n); we now briey indicate how to deduce the corresponding result for the lim inf: dene a sequence Can some explain the lim sup and lim inf? In my text book the definition of these two is this. Let (sn) be a sequence in \mathbbR. We define \lim \sup\ sn \limN \rightarrow \infty John Nachbar Washington University February 11, 2016 Lim Sup and Lim Inf.

Informally, for a sequence in R, the limit superior, or limsup, of a sequence is the Some fact about sup, inf, limsup and liminf 1 Supremum and Inmum For a set X of real numbers, the number supX, the supremum of X ( or least upper Show transcribed image text (2) For a set X and a sequence of subsets (Ek)kEN, with each Ek CX, define lim inf Ek an lim sup Ek: Take X R and define the sequence (ER) by setting, for each n EN, nand Identify (with proof) the sets lim inf Ek and lim sup The first is to change the sequence into a convergent one (extract subsequences) and the second is to modify our concept of limit (lim sup and lim inf).

Definition: Subsequence Let be a sequence. LIMINF and LIMSUP for bounded sequences of real numbers De nitions Let (c n)1 n1 be a bounded sequence of real numbers. De ne a n inffc k: k ng; and b n supfc k: k ng: The sequence (a n) is bounded and increasing, so it has a limit; call it a. This limit is by de nition the liminf of the sequence (c

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