Consider a continuous function $f: \mathbb{R} \times \mathbb{R}_{>0} \to \mathbb{R}_{\geq 0}$ such that $f_n=f(x,y_n)$ converges to zero uniformly over $x$ whenever $(y_n)$ approaches zero from above. That is, take a sequence $(y_n)$ such that $y_n \longrightarrow 0^+$, then $$ f_n = \sup_x f(x,y_n) \longrightarrow 0^+ $$ Consider a measure $\mu$ acting over $x$. Based on the dominated convergence theorem (DCT) for uniformly convergent sequences one has that for $$ \int f_n d\mu(x) \longrightarrow 0. $$
Now consider another function $h : \mathbb{R}_{>0} \rightarrow \mathbb{R}$ with $\lim_{y \to 0^+} h(y) = -\infty$. The function $h$ is continuous. In the paper that I am reading it is argued that now $$ \int h( f_n ) d\mu(x) \longrightarrow -\infty. $$ I can get the intuition why the latter should be correct. However, it is not clear to me how to justify it formally using DCTs. Let $(\theta_n)$ be the sequence $\theta_n = h(\sup_x f(x,y_n))$ where clearly $\theta_n \rightarrow -\infty$. How can I use the DCTs here to say that \begin{equation}\tag{1} \int \theta_n d\mu(x) \longrightarrow -\infty~? \end{equation} Usually in DCT one needs the sequence of function to converge to a finite limit. Am I wrong that, given the framework above, (1) cannot be justified?