Let $X_1, X_2, \ldots$ be i.i.d. with mean $\mu$ and variance $\sigma^2 < \infty$. Let $\bar{X}_n^b$ denote the mean of a bootstrap sample of size $n$ drawn from the empirical distribution $P_n$. Then, almost surely, \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\!\left(\frac{t}{\sigma}\right)\right| \to 0, \end{align*} where $\Phi$ is the standard normal c.d.f.