Convergence using Holder's inequality Assuming $X_n→X$ in $L^{p}$ and $Y_n→Y$ in $L^{q}$,
where $p > 1$ and $1/p + 1/q = 1$, I want to prove that $X_{n}Y_{n} \to XY$ in $L^{1}$.
I know I have to use Holder's inequality here, but I am stuck at how to begin.

$$E|X_nY_n-XY| \leq E|X_nY_n-X_nY|+E|X_nY-XY|$$ $$\leq \|X_n\|_p \|Y_n-Y\|_q+\|X_n-X\|_p \|Y_q\| \to 0.$$  Note that $X_n \to X$ in $L^{p}$ implies that $(\|X_n\|_p)$ is bounded. 
