I want to do : Consider two sequences $\{a_n\}_{n\in \mathbb{N}}\subset \mathbb{R},\{b_n\}_{n\in\mathbb{N}}\subset \mathbb{R}_+$.
We assume that :
- $(*)$ $\liminf_{n} a_n \geq A \in \mathbb{R}$
- $(**)$ $\lim b_n=B \in \mathbb{R}_+$
$\textbf{Claim :}$ $$ \liminf_n a_nb_n \geq AB. $$