 Solution

Let ${\epsilon }_1,\dots ,{\epsilon }_n$ be independent $\mathrm{Bernoulli}(\lambda /n)$ variables and $S_n={\sum }_i{\epsilon }_i$. Apply the tensorization of entropy to $f(S_n)$ and then apply the stated Bernoulli log-Sobolev inequality in each coordinate. If $S_n^{(i)}=S_n-{\epsilon }_i$, this givesFor each fixed $i$, $(S_n^{(i)},{\epsilon }_i)$ converges in distribution to $(X,0)$, where $X\sim \mathrm{Poisson}(\lambda )$. The Poisson limit theorem in fact gives convergence in total variation. The assumptions $K_1\le f\le K_2$ and $|Df|\le K_3$ make all displayed integrands bounded, so expectations and entropy pass to the limit. Since $n(\lambda /n)(1-\lambda /n)\to \lambda$,