Consider the functions $f_p(x):=\Vert x\Vert_p^p$. Clearly, $f_p\to f_1$ pointwise. The question is: does it converge also epigraphically? In synthesis, this means that the epigraphs $epi(f_p)$ (i.e. the sets $epi(f_p)=\{(x,\alpha)\in\mathbb{R}^n\times\mathbb{R} : f(x)\leq \alpha\}$) approach $epi(f_1)$ as $p\to1$, which is intuitively true). In this case, the epigraphical convergence of $f_p$ to $f$ is equivalent to the validity of the following conditions:
\begin{equation} \begin{cases} \liminf_{p\to1^+}f_p(x_p)\geq f_1(x) & \text{for every sequence $(x_p)_p$ converging to $x$},\\ \limsup_{p\to1^+}f_p(x_p)\leq f_1(x) & \text{for some sequence $(x_p)_p$ converging to $x$}. \end{cases} \end{equation}
These conditions seems to be holding trivially in this situation, but maybe I'm missing something.
Moreover, what can we say about the epigraphical convergence of $f_p(Lx)$ to $f_1(Lx)$ for some linear operator $L:\mathbb{R}^n\to\mathbb{R}^m$ (with an abuse of notation, now $f_p$ denotes the $\ell^p_m$ norm)?
As a reference, I'm using "Variational Analysis" by R. Tyrrell Rockafellar and Roger J. B. Wets (see Chapter 7).