Accumulation points of product sequence

Let $a_n$ and $b_n$ be two bounded real-valued sequences.

Then if $a$ is an accumulation point of $a_n$ and $b$ an accumulation point of $B$, it is clear that $ab$ is an accumulation point of $a_n b_n$. Indeed, there exist subsequences $a_{n_j}$ and $b_{n_k}$ converging to $a$ and $b$ respectively, and so (by taking further subsequences) we may assume these to be the same subsequence, hence the product $a_{n_j}b_{n_k}\to ab$.

It follows that if $A$, $B$, and $C$ are the sets of accumulation points of $a_n$, $b_n$, and $a_n b_n$ respectively, then

$$C\subseteq AB.$$

Question 1: Is the reverse inclusion $AB\subseteq C$ true?

Question 2: If the answer to Q1 is no, then does $AB\subseteq C$ hold when $a_n$ is convergent?

Thoughts: In the case that one of the sequences is allowed to be unbounded, even the first inclusion fails; say if $a_n=n$ and $b_n=1/n$, then $A=\emptyset$, $B=\{0\}$, so $AB=\emptyset$ but $C=\{1\}$.