Limit inferior and limit superior

Limit inferior and limit superior are mathematical concepts used to describe the behavior of a sequence. The limit inferior is the smallest number that the sequence approaches infinitely often, while the limit superior is the largest number that the sequence approaches infinitely often. These concepts are useful in analyzing the convergence or divergence of a sequence and can provide insight into the behavior of the sequence as a whole.From: Basic Analysis I [2020]

Preliminaries

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Published in James K. Peterson, Basic Analysis II, 2020

James K. Peterson

We will often be in situations where the classical limits fail to exist. However, even if this is the case, limit inferiors and limit superiors do exist! Let(an) be a sequence of real numbers, Let (an1) be any subsequence of (an) which converges. If (an) is a bounded sequence, the Bolzano - Weierstrass Theorem guarantees there is at least one such subsequence. Consider the set S={α∈ℜ|∃(an1)⊂(an)∋an1→α} The set S is called the set of subsequential limits of the sequence (an). We define the limit inferior and limit superior of a sequence using S.

A note on the marginal instability rates of two-dimensional linear cocycles

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Published in Dynamical Systems, 2023

Ian D. Morris, Jonah Varney

We remark that the situation described in Theorem 1.1 is quite delicate: if the dimension of the linear maps is raised from 2 to 3, or if a shift over a compact infinite alphabet is allowed in place of the finite alphabet , then the conclusion no longer holds and the above sequences may grow at a rate strictly intermediate between linear growth and boundedness (see Refs. [13,21,23]). In this article, we give an alternative proof of the above result which is due to the second named author and which was previously presented in the thesis [29]. We remark that the actual existence of the limit (2) is a new contribution originating in this article: in Refs. [13,29], it was shown that the limit inferior and limit superior of this sequence are finite and nonzero, but it was not shown that they are equal to one another.

Limit inferior and limit superior

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Preliminaries

A note on the marginal instability rates of two-dimensional linear cocycles