4. (10 分) 假設 \( X_1, ..., X_n \stackrel{iid}{\sim} N(\mu, \sigma^2) \) 且 \( \mu \in \mathbb{R} \& \sigma^2 \in (0, \infty) \)。令
\[ \overline{X}_n = \frac{1}{n} \sum_{i=1}^{n} X_i \]
\[ S_n^2 = \frac{1}{n-1} \sum_{i=1}^{n} (X_i - \overline{X}_n)^2 \]
試證 \( \frac{\sqrt{n}(\overline{X}_n - \mu)}{S_n} \xrightarrow{D} N(0,1) \) 當 \( n \to \infty \)。