Convergence of the expected value of bounded random variables

Question

Let $X_n$ be a sequence of bounded random variable such that $$ \mathbb{E}\left[X_n\right]\to\mathbb{E}\left[X\right] $$ with $X$ a bounded random variable. Can I conclude that

$$ \mathbb{E}\left[X_n\,W\right]\to\mathbb{E}\left[X\,W\right] $$

for any bounded random variable $W$ ?

Answer 1

No.

Throw a fair coin and for every $n$ let $X_n=1$ if it lands on heads and $X_n=0$ otherwise.

Let $X=1$ if it lands on tails and $X=0$ otherwise.

Then let $W=X$ so that $X_nW=0$ and $XW=X$.

Answer 2

Let $X$ have uniform distribution on $(-1,1)$, $X_n=-X$ for all $n$ and $W=X$. Then $EX_n W=-1/3$ for all $n$ and $EXW=1/3$ even though $EX_n =0 \to 0=EX$.