I am reading a paper in which the author wants to prove the convergence of the moments. He transforms the object of interest $\varepsilon^{-1} (\vartheta_\varepsilon^*-\vartheta_0)$ into \begin{align*} \varepsilon^{-1} (\vartheta_\varepsilon^*-\vartheta_0)=\Delta_\varepsilon + \frac{\bar{\vartheta}_{\tau_\varepsilon}-\vartheta_0}{\varepsilon} R_\varepsilon^* \end{align*} Now, he shows that $\Delta_\varepsilon$ converges uniformly to the normal distribution $N(0,I(\vartheta_0)^{-1})\overset{d}{=}:\zeta$ (using the uniform CLT) and that \begin{align*} \sup_{\vartheta_0 \in \mathbb{K}}\mathbb{E}_{\vartheta_0} \left| \frac{\bar{\vartheta}_{\tau_\varepsilon}-\vartheta_0}{\varepsilon} R_\varepsilon^* \right|^p \underset{\varepsilon \rightarrow 0}{\longrightarrow} 0 \end{align*}
Hence, $\varepsilon^{-1} (\vartheta_\varepsilon^*-\vartheta_0) \rightarrow N(0,I(\vartheta_0)^{-1})$ as $\varepsilon \rightarrow 0$. But he also claims that the moments converge (as a consequence of this proof!), i.e. that for any $p>0$: $\mathbb{E}_{\vartheta_0} |\varepsilon^{-1} (\vartheta_\varepsilon^*-\vartheta_0)|^p \rightarrow \mathbb{E}_{\vartheta_0}|\zeta|^p $.
My question: Why does this hold?
I am of course familiar with the well-known counterexamples of "convergence in distribution implies convergence of moments" and the ability to use uniform integrability. But here the author seems (as he claims so) to have shown the moment convergence already; but I am unable to see this connection.