If X$_n$ is dominated by some Y in L$^1$ or if it is identically distributed with finite mean, then prove that X$_n$ is uniformly integrable.
Approach:
Start with dominated convergence theorem but then how to use the fact of identically distributed with finite mean and move towards X$_n$ being uniformly integrable.
Any suggestions will be greatly appreciated.
Fix $K>0$ and note that if $|X_n|\leq Y$ then $$ E|X_n|I(|X_n|\geq K)\leq EYI(|X_n|\geq K) $$ for all $n$, whence $$ \sup_{n}E|X_n|I(|X_n|\geq K)\leq EYI(|X_n|\geq K)\to 0 $$ as $K\to \infty$ as desired.
The case of i.i.d with finite mean follows from the equality $$ E|X_n|I(|X_n|\geq K)= E|X_1|I(|X_1|\geq K) $$ and similar reasoning as above.