Let $Z_n$ be a variable that converges weakly in $L^1$ to $Z$, that is for all $E\in\mathcal{F}$ it holds that
$$ \mathbb{E}\left[Z_n\,I(E)\right]\to \mathbb{E}\left[Z\,I(E)\right]. $$
Let now $X$ be a bounded random variable. A want to prove that $Z_n\,X$ converges weakly to $Z\,X$, that is
$$ \mathbb{E}\left[(Z_n-Z)\,X\,I(E)\right]\to 0. $$
In order to do prove this I am trying to find a proper inequality, I thought to the Holder inequality but it does not seem to be the proper way:
$$ \left|\mathbb{E}\left[(Z_n-Z)\,X\,I(E)\right]\right|\leq \mathbb{E}\left[\left|(Z_n-Z)\,X\,I(E)\right|\right]\leq C\,\mathbb{E}\left[\left|(Z_n-Z)\,I(E)\right|^p\right]^{1/p} $$
with $p>1$.
You need a theorem from Functional Analysis which says that weakly convergent sequences are norm bounded. Hence $\sup \{\int |Z_n|+\int|Z| \}<\infty$. Now the hypothesis implies that $EZ_nY \to EZY$ for any simple function $Y$. Since $XI_E$ is a bounded measurable function there exists a simple function $Y$ such that $||XI_E-Y||_\infty<\epsilon$. Hence $E|Z_nXI_E-ZXI_E|\leq \epsilon \int |Z_n| +\epsilon\int |Z| +|\int Z_nY-\int ZY|$. Let $n \to \infty$ and then $\epsilon \to 0$.