Let $X$ be a real random variable and $(X_n)_n$ be a sequence of real random variable, such that : $$\exists \alpha>0; \lim_n\int_{\mathbb{R}}|F_{X_n}(x)-F_X(x)|^{\alpha}dx=0$$
I need to prove that $(X_n)_n$ converges weakly to $X$.
I am thankful for any idea.
Hint: follow the following steps:
1) there is a subsequence $F_{X_{n_j}}$ converging almost everywhere to $F_X$. Since complement of a set of measure $0$ is dense it follows that $F_{X_{n_j}}(x)$ converges to $F_X(x)$ for all $x$ in dense set.
2) Use standard argument to show that 1) and monotonicity of $F_X$ implies converges at every point $x$ where $F_X$ is continuous.
3) Conclude that $X_{n_j}$ converges weakly to $X$.
4) Apply the above result to subsequences of $F_{X_n}$ to conclude that the entire sequence $X_n$ converges weakly to $X$. [ Weak convergence is equivalent to the fact that $Ef(X_n) \to Ef(X)$ for every bounded continuous function. From this it follows that if every subsequence of $X_n$ has a further subsequence which converges weakly to $X$ then $X_n \to X$ weakly].