Definition: Limit Superior

Let $(a_n)_{n\in\mathbb N}$ be a real sequence. The limit superior $\varlimsup a_n$ is the element of the extended real numbers $\overline{\mathbb R}$ reached for $n\to\infty$ by the decreasing sequence $(\sup D_n)_{n\in\mathbb N}$, where $\sup D_n$ denotes the supremum of extended real numbers for the set $D_n:=\{a_k:~k\ge n\},$ formally

$$\varlimsup_{n\to\infty} a_n:=\lim_{n\to\infty}(\sup D_n).$$


This definition is motivated by the following facts: * The sequence $\sup D_n$ is monotonically decreasing. * This sequence is bounded below by definition. * Every bounded monotonic sequence is convergent. * Therefore, a limit $\lim_{n\to\infty}(\sup D_n)$ must definitely exist, if the sequence $(a_n)_{n\in\mathbb N}$ is bounded. * If the sequence is unbounded, it exists indefinitely by setting $$\varlimsup_{n\to\infty} a_n=+\infty.$$

Propositions: 1 2

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  1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983