Solution
= Solution
A <martingale> has <uniform integrability> when its entire family of time values has uniformly small integrable tails:
$$
\boxed{\lim_{K\to\infty}\sup_{n\geq0}\mathbb E\bigl[|X_n|\mathbf1_{\{|X_n|>K\}}\bigr]=0.}
$$
Here every $X_n$ is an <integrable random variable>. This implies $\sup_n\mathbb E|X_n|<\infty$, because the contribution from $|X_n|\leq K$ is at most $K$. Mere boundedness of these <expectations> does not imply <uniform integrability>.