Solution
= Solution
Uniform integrability means
$$
\lim_{K\to\infty}\sup_n
\mathbb E\bigl[|X_n|\mathbf1_{\{|X_n|>K\}}\bigr]=0.
$$
If $\sup_n\mathbb E|X_n|^p=C<\infty$ for $p>1$, then
$$
\mathbb E[|X_n|\mathbf1_{|X_n|>K}]
\leq K^{1-p}\mathbb E|X_n|^p\leq CK^{1-p},
$$
proving uniform integrability. For a counterexample, let $X_n=n$ with probability $1/n$ and zero otherwise. Then $\mathbb E|X_n|=1$, but for every $K$ and $n>K$ the tail expectation is one.