I am trying to prove (or disprove) this proposition:
Let $(X_{n})_{n \ge 0}$ and $(Y_{n})_{n \ge 0}$ be two sequences of random variables, defined on some probability space $(\Omega, \mathcal{F}, \mathbb{P})$. Assume that:
1-. $X_{n}$ and $Y_{n}$ are non-negative sequences bounded above by $1$.
2-. $(X_{n})_{n \ge 0}$ converges a.s. to a constant $c > 0$.
3-. $\lim_{n \to \infty }\mathbb{E}(Y_{n})=d$.
Under the above conditions: is it true that $\lim_{n \to \infty}\mathbb{E}(X_{n}Y_{n})=cd$?
Observe that if $d=0$, we would have: \begin{align} \mathbb{E}(X_{n}Y_{n})\le \mathbb{E}(Y_{n}) \end{align}
because $X_{n}$ is bounded above; and so, in this case the result is easy.
But, what happens when $d >0$? Does the proposition hold in the general case? And, if not, can anyone provide a counter-example?