= Solution
A sequence of <integrable random variables> $(X_n)$ has <uniform integrability> exactly when
$$
\boxed{\lim_{K\to\infty}\sup_{n\geq0}\mathbb E\left[|X_n|\mathbf1_{\{|X_n|>K\}}\right]=0.}
$$
The <uniformly integrable martingale convergence theorem> states that a <uniformly integrable> discrete-time <martingale> $(M_n)$ has an <integrable random variable> $M_\infty$ with
$$
\boxed{M_n\to M_\infty\ \text{almost surely and in }L^1,\qquad M_n=\mathbb E[M_\infty\mid\mathcal F_n].}
$$
Indeed, <uniform integrability> implies $\sup_n\mathbb E|M_n|<\infty$, so the <martingale convergence theorem> gives <almost sure convergence>. The combination of <uniform integrability> and <almost sure convergence> gives <convergence in L1>. For $m\geq n$, the <martingale> identity $\mathbb E[M_m\mid\mathcal F_n]=M_n$ passes to the limit by the <L1 contraction of conditional expectation>.
Conversely, <uniform integrability of conditional expectations> shows that a <martingale> of the form $M_n=\mathbb E[Y\mid\mathcal F_n]$ for one <integrable random variable> $Y$ is <uniformly integrable>.
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