We defined that $X_n\rightarrow X$, i.e, that $X_n$ converges in quadratic mean to $X$, if $\mathbb{E}[(X_n-X)^2]\rightarrow0$ as $n\rightarrow\infty$. I'm trying to prove the following statement: $$X_n\rightarrow X, Y_n\rightarrow Y \Rightarrow X_n+Y_n\rightarrow X+Y$$
I did the following:
\begin{split}
\mathbb{E}[(X_n+Y_n-(X+Y))^2] &= \mathbb{E}[(X_n-X+Y_n-Y)^2] \\
&= \mathbb{E}[(X_n-X)^2]+2\mathbb{E}[(X_n-X)(Y_n-Y)]+\mathbb{E}[(Y_n-Y)^2] \\
\end{split}
I know that the first and third terms go to $0$ as $n\rightarrow \infty$, but what happens to the term in the middle? How do I show that it goes to $0$ as $n\rightarrow \infty$?