Let $X_1, X_2, \ldots$ be i.i.d. from $P_{\theta_0}$ and suppose the model satisfies Assumption 2.1. Then as $n \to \infty$,
\begin{align*}
\sup_{\theta \in \Theta} \left| \bar{\ell}_n(\theta) - \ell(\theta) \right| \xrightarrow{\mathbb{P}} 0.
\end{align*}