Convergence in distribution of product of random variable

Question

Assume that $X$ is a bounded random variable, and $Y_n$ convergence in distribution to $Y$. All the moments of $Y_n$ and $Y$ exist. Is it true that $$ XY_n\to XY\quad in \quad distribution? $$

Answer 1

No. Let $X=Y$ and $Y_n=-Y$ for all $n$ where $X$ is bounded and has a symmetric distribution, say uniform distribution on $(-1,1)$. Then $Y_n \to Y$ in distribution because the distribution is symmetric but $XY_n=-X^{2}$ does not tend to $XY=X^{2}$ in distribution.