# Lim Sup And Lim Inf Pdf

We have max(S) = lim sup sn and min(S) = lim inf sn. Proof. We only show the first equality. In order to simplify notation we put L:= limsup sn. supan inf n bn. If every bn, then we define lim n supan. (2) If every cn R, then sup cn: n N is called the lower limit of an , denoted by. LIMINF and LIMSUP for bounded sequences of real numbers. Definitions. Let (cn). ∞ n=1 be a bounded sequence of real numbers. Define an = inf{ck: k ≥ n}. LIM INF AND LIM SUP. JOHN QUIGG. Definition. Let (xn) be a bounded sequence in R. The lim sup of (xn) is lim supxn:= lim k→∞ sup{xn: n ≥ k}, and the lim. We begin by stating explicitly some immediate properties of the sup and inf, which we use below. Proposition 1. (a) If A ⊂ R is a nonempty set, then inf A ≤ sup.

Lim Sup and Lim Inf. Informally, for a sequence in R, the limit superior, or limsup, of a sequence is the largest subsequential limit. Although I could use this as the de nition of limsup, the following alternate characterization, which does not even need a metric, turns. That is, lim n infan sup n cn. If every cn, then we define lim n infan. Remark The concept of lower limit and upper limit first appear in the book (AnalyseAlge’brique) written by Cauchy in But until , Paul du Bois-Reymond gave explanations on them, it becomes well-known. 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 .

## Limit Superior and Limit Inferior

xn:= inf{y ∈ R: xn < y for infinitely many n ∈ N}. Our hope is that lim sup n→∞ xn = lim n→. Handout: Examples of lim sup and lim inf. Example Calculate lim sup an and lim inf an for an = (−1)n(n + 5)/n. Solution Define αn = sup {ak|k ≥ n}. Then αn. This completes that proof that limsup an is a cluster point. Note that we have proven that an → −∞ if lim sup an = −∞. The corresponding statement for lim inf is that. In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting Whenever lim inf xn and lim sup xn both exist, we have. lim inf n → ∞ x n ≤ lim sup n → ∞ x n. {\displaystyle \liminf _{n\to \infty }x_{n}\leq \limsup _{n\to \infty. LIM SUP AND LIM INF. 1. Definitions and Theorems. Definition Let 1anl be a sequence of real numbers. A real num- ber l ∈ R is called a cluster point of.