Suppose that for a cdf $F(x)$ related to $X$ and its sample counterpart $F_n(x)$ we have,
$a_n = \sup _x |F_n(x) - F(x)| = o(1), x \in \mathbb{R}$
Then, for every continuous function such that $|f(x)| < K$ for every $x$ then,
$$|E(f(X_n) - E(f(X)| = o(1)$$
Does my proof is correct?
$$| \int_\mathbb{R} f(x) dF_n(x) - \int_\mathbb{R} f(x) dF(x)| \leq \int |f(x)|d(F_n(x) - F(x)) \leq K \int_\mathbb{R} 1d(F_n(x) - F(x))$$
Then, by integration by parts
$$\int_\mathbb{R} 1d(F_n(x) - F(x)) = \lim _{x \rightarrow \infty} (F_n(x) - F(x)) - \lim _{x \rightarrow - \infty} (F_n(x) - F(x)) - 0 \leq 2 a_n = o(1)$$
Since $\int_\mathbb{R} 0(F_n(x) - F(x)) dx = 0 $