The question is presented in

a dominated convergence theorem for martingale

Let $\{(X_1^n, X_2^n)\}_n$ be a sequence of martingales defined some probability space. (which means $E[X_2^n|X_1^n]=X_1^n$)

Suppose there exists $(X_1, X_2)$ such that

\begin{eqnarray} X_1^n&\to& X_1,~ a.s \\ X_2^n&\to& X_2,~ a.s \end{eqnarray}

and

\begin{eqnarray} \lim_{n\to\infty}E[|X^n_2|]=E[|X_2|] \end{eqnarray}

Can we prove

$$\lim_{n\to\infty}E[|X_1^n-X_1|]=0$$