Basic Properties of limsup and liminf

Let $(a_n)$ and $(b_n)$ be real sequences. Their limit superior and limit inferior, regarded as extended real numbers, satisfy:

1. $\liminf_{n\to\infty}a_n\le \limsup_{n\to\infty}a_n$.
2. If $a_n\le b_n$ eventually, then
   $$\limsup a_n\le\limsup b_n \quad\text{and}\quad \liminf a_n\le\liminf b_n.$$
3. $\liminf a_n=-\limsup(-a_n)$.
4. If $a_n\to L\in\mathbb R$, then $\limsup a_n=\liminf a_n=L$.
5. If $\limsup a_n<\alpha$, then $a_n<\alpha$ eventually. If $\liminf a_n>\beta$, then $a_n>\beta$ eventually.