Suppose
(1) $X_n \xrightarrow{d} X$
(2) $h: \mathbb{R} \rightarrow \mathbb{R}^+$ is continuous with $sup_{n \in \mathbb{N}}E(h(X_n)) < \infty$
(3) $g:\mathbb{R} \rightarrow \mathbb{R}$ is continuous and $g(x)/h(x) \rightarrow 0$ for $|x| \rightarrow \infty$
I want to prove $E(g(X_n)) \rightarrow E(g(X))$ as $n \rightarrow \infty$ but I don't really know where to start.
My initial idea is to say that $g(x)/h(x)$ is bounded and continuous by the continuity of $g$ and $f$ so that $E(g(X_n)/h(X_n)) \rightarrow E(g(X)/h(X))$ but I think this is wrong since I don't know where to go from there.
How do I go about this? Any hint/help is much appreciated. Thanks in advance!