Let $(X_n,\mathcal F_n)$ be a martingale bounded from below i.e. $X_n\geq M$ for some $M\in\mathbb R$. Then show that $\sup_n E|X_n|<\infty$.
It is easy to observe that $X_n$ converges almost surely to some $X\in L^1$ as $X_n-M$ is a non-negative martingale. I can conclude $X_n$ is $L^1-$bounded if I can show $X_n$ is uniformly integrable. But I don't know how to prove it.
$X_n^{-} \leq M^{-}$ so $EX_n^{-}$ is bounded. Since $X_n=EX_1$ for all $n$ (by martinagle property) we also know that $EX_n^{+}-EX_n^{-}=EX_n$ is bounded. This makes $E|X_n|=EX_n^{+}+EX_n^{-}$ bounded.