Let $\mathbb P_\theta=U[0,\theta]$.
For $h,\theta_0>0$ and $Z\sim\mathrm{exp}\left(\frac{1}{\theta_0}\right)$ I have to show that: $$\frac{\mathrm d\mathbb P^n_{\theta_0-h/n}}{\mathrm d\mathbb P^n_{\theta_0}}\overset{d,\mathbb P^n_{\theta_0}}{\longrightarrow} e^\frac h {\theta_0}\mathbb 1_{\{Z\geq h\}}$$
I already proved that for $Z_n=n(\theta_0-\max\{X_1,\dots,X_n\})$ with $X_1,\dots X_n\sim \mathbb P_{\theta_0}$ holds $Z_n\overset{D}{\to}Z$ and this task seems like I have to prove that the pdf is converging too. I'm not sure which technical steps I need to show this and I'm not sure which kind of convergence is meant by $d,\mathbb P^n_{\theta_0}$.
The convergence follows with Radon-Nikodym, Slutzky and continuous mapping theorem. \begin{align*} \frac{\mathrm d\mathbb P^n_{\theta_0-h/n}}{\mathrm d\mathbb P^n_{\theta_0}}&=\frac{\mathrm d\mathbb P^n_{\theta_0-h/n}}{\mathrm d\lambda}\left(\frac{\mathrm d\mathbb P^n_{\theta_0}}{\mathrm d\lambda}\right)^{-1}\\ &=\left(\frac{\theta_{0}}{\theta_{0}-\frac{h}{n}}\right)^{n}\frac{\mathrm 1_{\{0\leq X_i\leq\theta_0-h/n\ \forall i\}}}{\mathrm 1_{\{0\leq X_i\leq\theta_0\ \forall i\}}} \\ &=\left(\frac{\theta_{0}}{\theta_{0}-\frac{h}{n}}\right)^{n}\mathrm 1_{\{Z_n\geq h\}}\\ &\overset{\text{Slutzky}}{\to} \mathrm{exp}\left(\frac h {\theta_0}\right)\mathrm 1_{\{Z\geq h\}} \end{align*}