Let $(a_{n})_{n\in\mathbb{N}}\subseteq \bar{\mathbb{R}}$. The liminf and limsup of $(a_{n})_{n\in\mathbb{N}}$ have the following properties: 1. $$\liminf_{n\ge 1}a_{n}\le\limsup_{n\ge 1}a_{n}$$ 2. $$\limsup_{n\ge 1}(-a_{n})=-\liminf_{n\ge 1}a_{n}$$ 3. $$\limsup_{n\ge 1}a_{n}=\liminf_{n\ge 1}a_{n}\implies \lim_{ n \to \infty } a_{n}\text{ exists and equals both}$$ 4. $$\limsup_{n\ge 1}(a_{n}+\alpha_{n})\le \limsup_{n\ge 1}a_{n}+\limsup_{n\ge 1}\alpha_{n}$$iff we don’t encounter $\infty-\infty$ or $-\infty+\infty$ 5. If $a_{n}\le \alpha_{n},\forall n\ge 1$ then $$\begin{align*} &\liminf_{n\ge 1}a_{n}\le\liminf_{n\ge 1}\alpha_{n}\\ &\limsup_{n\ge 1}a_{n}\le\limsup_{n\ge 1}\alpha_{n} \end{align*}$$