This question is extended from Resnick's exercise 5.13 in his book A Probability Path.
Let the probability space be the Lebesgue interval, that is, $(\Omega=[0,1],\mathcal{B}([0,1]),\lambda)$ and define $X_n:=\frac{n}{\log n}1_{(0,\frac 1n)}$.
Show $X_n\to 0, E(X_n)\to 0$ even though DCT fails. And secondly, show
$\lim_{M\to\infty} \sup_{n\ge 2} E(X_n 1_{X_n>M})=0$ (uniform integrability)