Let $X_1, X_2, \ldots$ be i.i.d. draws from a distribution $P$ with mean $\mu$ and finite variance $\sigma^2 > 0$. Then as $n \to \infty$, \begin{align*} \sup_{t \in \mathbb{R}} \left|P_n\!\left(\sqrt{n}(\bar{X}_n^b - \bar{X}_n) \leq t \,\Big|\, X_1, \ldots, X_n\right) - \Phi_{\sigma^2}(t)\right| \xrightarrow{a.s.} 0, \end{align*} where $\Phi_{\sigma^2}$ denotes the cumulative distribution function of the $N(0, \sigma^2)$ distribution.