Can we always conclude limit superior and limit inferior of $\langle a_n\rangle=\langle b_nc_n\rangle$?

Question

Suppose that $\langle b_n\rangle$ and $\langle c_n\rangle$ be two sequences which are bounded and we know their limit superiors and limit inferiors.If we define a new sequence $\langle a_n\rangle$ as $\langle a_n\rangle=\langle b_nc_n\rangle$. My question is "Can we always conclude limit superior and limit inferior of $\langle a_n\rangle$?

Answer 1

No. Suppose $b_{2n}=1, b_{2n+1}=0, c_{2n}=0, c_{2n+1}=-1 $. Then $b_nc_n = 0$ for all $n$.