Uniform integrability, convergence in probability and weak convergence. Suppose that $T_n$ is a sequence of random variables defined on the probability space $(\Omega,\mathcal{F},\mathbb{P})$ and suppose that $T_n$ weakly converges to $1$, that is 
$$
\mathbb{E}[T_n\,I(E)]\to \mathbb{E}[I(E)]
$$
for all $E\in\mathcal{F}$. Let $W_n$ be a sequence of random variables such that $W_n\stackrel{p}{\to} 0$. Suppose that $T_n\,W_n$ is uniformly integrable.
I need to prove 
$$
\mathbb{E}[T_n\,W_n\,I(E)]\to 0
$$
for all $E\in\mathcal{F}$. I think this is a pretty known result but I miss a valid reference. 
 A: Assume that $W_n\to 0$ almost surely. Then by Egoroff's theorem, for each positive $\delta$, there exists a set $A$ such that $\mathbb P\left(\Omega\setminus A\right)\lt \delta$ and $s_n:=\sup_{\omega \in A}\left\lvert W_n\left(\omega\right)\right\rvert \to 0$. Then 
\begin{align}
\left\lvert \mathbb{E}\left[T_n\,W_n\,I(E)\right]\right\rvert &\leqslant 
\left\lvert \mathbb{E}\left[T_n\,W_n\,I\left(E\cap A\right)\right]\right\rvert+\left\lvert \mathbb{E}\left[T_n\,W_n\,I\left(E\cap A^c\right)\right]\right\rvert
\\ 
&\leqslant s_n\mathbb{E}\left[\left\lvert T_n I\left(E\cap A\right)\right\rvert\right]+\sup_{S\subset \Omega:\mathbb P(S)\leqslant \delta}\mathbb{E}\left[\left\lvert T_n\,W_n\right\rvert\,I\left(S\right)\right]\\
&\leqslant s_n\mathbb{E}\left[\left\lvert T_n  \right\rvert\right]+\sup_{S\subset \Omega:\mathbb P(S)\leqslant \delta}\mathbb{E}\left[\left\lvert T_n\,W_n\right\rvert\,I\left(S\right)\right].
\end{align}A sequence which converges weakly in $\mathbb L^1$ is bounded hence 
$$
\limsup_{n\to +\infty}\left\lvert \mathbb{E}\left[T_n\,W_n\,I(E)\right]\right\rvert\leqslant  \sup_{S\subset \Omega:\mathbb P(S)\leqslant \delta}\sup_{N\geqslant 1}\mathbb{E}\left[\left\lvert T_NW_N\right\rvert\,I\left(S\right)\right]
$$
and the last term can be made arbitrarily small by uniform integrability.
Assume now that $Y_n\to 0$ in probability.
Let $a_n:=\left\lvert\mathbb{E}[T_n\,W_n\,I(E)]\right\rvert$. If $a_n$ does not converge to $0$, then there exists a positive $\varepsilon$ and an increasing sequence of integers $\left(n_k\right)_{k\geqslant 1}$ such that $a_{n_k}\gt \varepsilon$ for all $k$. Extract from $\left(W_{n_k}\right)$ an almost surely convergent subsequence $\left(W_{m_N}\right)_{N\geqslant 1}$ and apply the previous part to the setting $\widetilde{W_N}:=W_{n_M}$, $\widetilde{T_N}:=T_{n_M}$ to find a contradiction.
