Suppose $g_{n}(\cdot)$ defined on $[0,1]$ converges in distribution to a continuous Gaussian process. Let $U_{1},...,U_{n}$ be i.i.d. random variables following $\text{Unif}[0,1]$. Allow $g_{n}$ to be correlated with the $U_{i}$s; for instance, $g_{n}(x)=\frac{1}{\sqrt{n}}\sum_{i}1(U_{i}\leq x)$. Is the following statement correct? $$\frac{1}{n}\sum_{i}g_{n}(U_{i}) \text{ has the same asymptotic distribution as } \int_{[0,1]} g_{n}(u)du.$$
I feel the statement is true but I'm not sure if my proof is correct: Let $\mu_{n}$ be the empirical c.d.f. for $U_{i}$. We can rewrite the average $\frac{1}{n}\sum_{i}g_{n}(U_{i})$ as $\int_{[0,1]}g_{n}(u)\mu_{n}(du)$.Hence, the difference of the considered two terms is $$\int_{[0,1]} g_{n}(u)(\mu_{n}-\mu)(du)\equiv f(g_{n},\mu_{n}-\mu).$$ By Glivenko-Cantelli, $\mu_{n}\to \mu$ almost surely where $\mu$ is the c.d.f. of $\text{Unif}[0,1]$. By convergence in distribution of $g_{n}(u)$ and by continuity of the functional $f$, the continuous mapping theorem implies that $f(g_{n},\mu_{n}-\mu)$ converges to 0 in probability.
Is there any error in my proof? Thank you very much.