 Solution

A martingale has uniform integrability when its entire family of time values has uniformly small integrable tails:Here every $X_n$ is an integrable random variable. This implies ${\sup }_n\mathbb{E}|X_n|<\infty$, because the contribution from $|X_n|\le K$ is at most $K$. Mere boundedness of these expectations does not imply uniform integrability.