Consider a random vector $\eta_n\equiv (\eta_1,..., \eta_n)$ with probability density function (pdf) $f_n$ and a random vector $D_n\equiv (D_1,..., D_n)$.
Let $f_{d_n}$ be the pdf of $\eta_n$ conditional on $D_n=d_n$.
I am reading a book where this claim is made and I have difficulties in interpreting the claim:
$$ \lim_{n\rightarrow \infty} \Big|\frac{f_{D_n}(\eta_n)}{f_n(\eta_n)}-1 \Big|=0 \text{ with probability approaching $1$ as $n\rightarrow \infty$} $$
I am very confused on the "double limiting": firstly a "traditional limit" with a random object ($\frac{f_{D_n}(\eta_n)}{f_n(\eta_n)}$ is a random variable because of $\eta_n$ and $D_n$) and secondly a limit of a sequence of probabilities. Any suggestion?