Suppose $X_n$, $n=1,2,... $ and $X$ are random variables defined on probability space $(\Omega,\mathcal{F},\mathbb{ P } )$. They have bounded first moments. Assume that $X_n \geq 0$ almost surely, $\mathbb{E}X_n = 1$ and $\mathbb{E}(X_n \log X_n) \leq 1$. Assume that for every bounded random variable $Y$, $\mathbb{E}(X_nY)\rightarrow \mathbb{E}(XY) $ as $n \rightarrow \infty$. Show that: $X_n \rightarrow X$ in probability and $\mathbb{E}(X\log X) \leq 1$.
In this problem, we know that $X_n,X \in L^{1}(\Omega,\mathcal{F},\mathbb{ P })$, and $X_n$ converges weakly to $X$ in $L^{1}(\Omega,\mathcal{F},\mathbb{ P })$, we need to deduce that $X_n$ converges in probability. Currently, I have no idea of this problem. Could anyone give me some hints?