In the paper: Whittle estimation for stationary state space models with finite second moments they have the following lemma:
Let $\Theta$ be a compact parameter space and let $g:[-\pi, \pi] \times \Theta \rightarrow \mathbb{R}$ be differentiable in the first component. Assume further that $\frac{\partial}{\partial \omega} g(\omega, \vartheta)$ is continuous on $[-\pi, \pi] \times \Theta$. Then, $$ sup_{\vartheta \in \Theta} | \frac{1}{2 n} \sum_{j=-n+1}^n g\left(\omega_j, \vartheta\right)-\frac{1}{2 \pi} \int_{-\pi}^\pi g(\omega, \vartheta) d \omega | \stackrel{n \rightarrow \infty}{\longrightarrow} 0 $$
Proof. Follows by an application of the mean value theorem.
I think that there is a typo: $g$ should be differentiable in the second component and not the first one. On top of that, I am not able see why the mean value theorem proves the uniform convergence in that specific case.