I have I have a collection of bounded variation and right-continuous functions, $(F_n)_{n \in \mathbb{N}}$, and another bounded variation and right-continuous function, $G$, which satisfy $$\sup_x \lvert F_n(x) - G(x)\rvert \to 0 \quad \text{as} \quad n \to \infty$$ In other words, $F_n$ converges to $G$ uniformly on the real line.
What I want to show is $$\left|\int_{x\in (-M,M)}x^2\,dF_n(x)-\int_{x\in (-M,M)}x^2\,dG(x)\right| \to 0$$
This seems intuitively true but how do I actually show it?