Uniform integrability from an Lp bound ($p>1$)

For $p>1$, ${\sup }_n\mathbb{E}|X_n{|}^p=C<\infty$ gives ${\sup }_n\mathbb{E}[|X_n|1_{|X_n|>M}]\le C{M}^{1-p}\to 0$. Thus an $L^p$ bound implies uniform integrability; a mere $L^1$ bound does not.