For $f$ continuous and bounded find $\mathbb{E} \big [ \prod_{i=1}^n f \big (\ X_i \big ) \big ]$ for random variables $X_1, X_2, \ldots, X_n$
I think this could go like this. We have $$\sqrt{n}\,Y^{(n)}_i=\frac{X_i}{\bigg(\frac{R_n}{\sqrt{n}}\bigg)}\stackrel{\textrm{a.s.}}\to X_i=Y_i$$ and by portmanteau theorem we get $$E[g_K(Y_n)]\to E[g_K(Y)]=E[f(X_1)f(X_2)(...)f(X_K)]=E[f(X_1)]^K$$ beacause $g(x)=\prod_{j=1}^Kf(x_j)$ is bounded and continuous.