 Moving-variable conditional-expectation convergence

If bounded random variables $X_n$ converge almost surely to $X$, thenalmost surely and in $L^1$. For $Z_n={\sup }_{m\ge n}|X_m-X|$, the variables $\mathbb{E}[Z_n\mid {\mathcal{F}}_n]$ form a nonnegative supermartingale with expectations tending to zero.