Expected Value of converging Random Variables
Use the monotone convergence theorem, i.e., $$ \mathsf{E}|X|=\mathsf{E}\lim_{k\to\infty}f_k(X)=\lim_{k\to\infty}\mathsf{E}f_k(X)\le \liminf_{n\to\infty}\mathsf{E}|X_n|. $$
Expected Value of converging Random Variables
Use the monotone convergence theorem, i.e., $$ \mathsf{E}|X|=\mathsf{E}\lim_{k\to\infty}f_k(X)=\lim_{k\to\infty}\mathsf{E}f_k(X)\le \liminf_{n\to\infty}\mathsf{E}|X_n|. $$