I am trying to understand the proof of the entropic central limit theorem presented here, but I am stuck on the step where the convergence in distribution is upgraded to convergence of the probability densities.
Let $S_n=\frac{1}{\sqrt{n}}\sum_{i=1}^nX_i$ where the $X_i$ are iid with $\mathbb{E}X=0$ and $\mathbb{E}X^2=1$. Then the random variable $S_n^u$ is defined to be $$e^{-u}S_n+\sqrt{1-e^{-2u}}Z$$ where $Z$ is a standard normal variable independent of $S_n$. The proof linked above establishes that some subsequence $S_{n_k}$ is convergent in distribution. Let $W$ be a random variable which has this limit distribution. It is then claimed that for $u>0$, the densities of $S_{n_k}^u$ converge pointwise to the density of $W$. The author claims that this follows from the smoothing introduced by the convolution, but I am stuck trying to put together the details.
My first thought was to use the pointwise convergence of the cumulative distribution functions $F^u_{n_k}$ established by the convergence in distribution and try to show that the densities $f^u_{n_k}$ are uniformly convergent, since then $$f^u_{n_k} = (F^u_{n_k})^\prime\to F^\prime=f$$ where $F$ and $f$ are the cdf and density of the standard normal, respectively.
To this end, I thought of perhaps extending the densities to the one-point compactification of $\mathbb{R}$ by simply setting $f^u_{n_k}(\infty)=0$. Then I was hoping to use some of the smoothness properties of the densities to apply Arzelà–Ascoli and establish the uniform convergence. However, I am not sure how to check the necessary uniform boundedness conditions in this case.
Any help with filling in the details of this part of the proof, or references to the kinds of results that would be used to establish claims like this, would be greatly appreciated.