Definition

Let $(a_{n})_{n\in\mathbb{N}}\subseteq \bar{\mathbb{R}}$. Define a new sequence $(c_{n})_{n\in\mathbb{N}}$ as $$c_{n}=\inf_{k\ge n}a_{k}$$where we observe $c_{1}\le c_{2}\le\dots$ (i.e. monotonically increasing). Then we define the limit inferior of our sequence as $$\liminf_{n\ge 1}a_{n}:=\sup_{n\ge 1}c_{n}=\sup_{n\ge 1}\inf_{k\ge n}a_{k}$$Let $T=\{ t\in \bar{\mathbb{R}}: \exists(a_{n_{k}})_{k\in\mathbb{N}}\subset(a_{n})_{n\in\mathbb{N}}:\lim_{ k \to \infty }a_{n_{k}}=t \}$. Then $$\limsup_{ n \ge 1 }a_{n}=\sup_{t\in T}t$$

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The limit inferior of $(a_n)$, $$\liminf_{n\to\infty}a_n$$ is defined as follows: 1. If $(a_n)$ is bounded below and $T\not=\emptyset$ then $\liminf_{n\to\infty}a_n$ is the smallest number in $T$ or $$\liminf_{n\to\infty}a_n=\inf T$$ 2. If $(a_n)$ is not bounded below then $$\liminf_{n\to\infty}a_n=-\infty$$ 3. If $(a_n)$ is bounded below and $T=\emptyset$ then $$\liminf_{n\to\infty}a_n=\infty$$