Is liminf of a product equal to the product of liminfs?

Question

My question is just for curiosity. I was thinking if is true this curious affirmation:

Let $a_n$ a bounded sequence of nonnegative numbers and $b_n$ a convergent sequence of negative numbers. Then $\lim \inf (a_n b_n) = (\lim \inf a_n) (\lim \inf b_n) $ This is true?

Answer 1

Let $b_n=-1$ for all $n$. Say $a_n$ alternates between 1 and 3. Then $(\lim\inf b_n)(\lim\inf a_n)=-1$, but $\lim\inf a_n b_n=-3$.