I have some doubts if this problem is right. Show that if $X,Y,X_n$ are r.v. in $L_2(\Omega,\mathcal{F},P)$ such that $X_n \xrightarrow{L_2} X$, then
$$X_nY \xrightarrow{L_1} XY$$
First of all, I need to prove that $E|X_nY-XY| \to 0$. Then,
$$E|X_nY-XY|=E|(Y)(X_n-X)| \leq \sqrt{E|Y|^2 E|X_n-X|^2},$$
since $X,Y,X_n$ r.v. in $L_2(\Omega,\mathcal{F},P)$ and Schwarz inequality. But $X_n \xrightarrow{L_2} X$ and hence my proof is done. Is it correct? I think it is too easy. If is not correct any help?