Let $\{f(\cdot, \theta) : \theta \in \Theta\}$ be a parametric model with $\Theta \subseteq \mathbb{R}$ satisfying the regularity assumptions of the asymptotic normality theorem (Assumptions 2.2 of the course). Let $\pi$ be a prior with continuous positive density at $\theta_0$, i.e.\ $\pi(\theta_0) > 0$. Let $\Pi_n = \Pi(\cdot \mid X_1, \ldots, X_n)$ be the posterior distribution, and let $\phi_n = \mathcal{N}(\hat{\theta}_n, I(\theta_0)^{-1}/n)$ be the Gaussian distribution centered at the MLE with variance equal to the asymptotic variance of the MLE. Then, under $X_i \overset{\text{i.i.d.}}{\sim} f(\cdot, \theta_0)$, \begin{align*} \|\Pi_n - \phi_n\|_{\mathrm{TV}} = \int_\Theta |\Pi_n(\theta) - \phi_n(\theta)|\, d\theta \xrightarrow{a.s.} 0 \quad \text{as } n \to \infty. \end{align*}