# Expected Value of Product of Sequences of Random Variables

I am trying to prove the following:

Let $X_{n}$ be a sequence of random variables converging in probability to some random variable $X$. Furthermore $P(|Xn|>k)=0$ for all n and some $k>0$.

Let $Y_{n}$ be a sequence in $L^{1}(\Omega)$. Assume that there exists a real number $\lambda$ such that $E(Y_{n}) = \lambda$ for all n and $\sup |Y_{n}| \le \eta$ for some $\eta \in L^{1}(\Omega)$.

Prove that if $X=c$ then $lim_{n \to \infty} E(Y_{n}X_{n})=c \lambda$.

How do I show this if the sequences are not independent?

• Are there any other assumptions about $X_n$'s (e.g. bounded or u.i.)? Even if $Y_n$'s are independent of $X_n$'s, there is no guarantee that $\mathbb{E}[X_n]\to c$. – d.k.o. Nov 7 '16 at 4:14