2. A sequence of random variables $\{X_n, n \ge 1\}$ is called uniformly integrable if for every $\varepsilon > 0$, there corresponds a $\delta > 0$ such that

$$\sup{n \ge 1} \intA |X_n| \, dP < \varepsilon,$$

whenever $P(A) < \delta$, and, in addition,

$$\sup{n \ge 1} E|Xn| < \infty.$$

Show that if $\{Xn, n \ge 1\}$ is uniformly integrable, then $\{Sn/n, n \ge 1\}$ is also uniformly integrable, where $Sn = \sum{i=1}^n X_i$. (20%)