Let \(\left(X, \mathscr{A}, \mathscr{A}_{n}, \muight)\) be a finite filtered measure space. Let \(u_{n} \in L^{1}\left(\mathscr{A}_{n}ight)\) be a sequence of measurable functions such that \(\left|u_{n}ight| \leqslant \mathbb{E}^{\mathscr{A}_{n}} f\) for some \(f \in L^{1}(\mathscr{A})\). Show that the family \(\left(u_{n}ight)_{n \in \mathbb{N}}\) is uniformly integrable.