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 \[ \sup_{n \ge 1} \int_A |X_n| dP < \varepsilon, \] whenever \( P(A) < \delta \), and, in addition, \[ \sup_{n \ge 1} E|X_n| < \infty. \] Show that if \( \{X_n, n \ge 1\} \) is uniformly integrable, then \( \{S_n/n, n \ge 1\} \) is also uniformly integrable, where \( S_n = \sum_{i=1}^n X_i \). (20%)