The limit of an integral using integrating by part

Question

Let $G(x)$ be the density of $N(0,1)$ random variable. Let $h$ be a function in $C^1(\mathbb{R})$ such that $h$ and $h'$ are both bounded. By integration by part, we have $$ \int_{-\infty}^{+\infty} G(x)(h'(x) - xh(x))\ dx = 0 $$ Let $g_n$ be a sequence of piecewise continuous functions such that for every $n$, we have $$\int_{-\infty}^{+\infty} g_n(x)\ dx= 1$$ and for every $h$ in $C^1(\mathbb{R})$ such that $h$ and $h'$ are both bounded, we have $$\lim_{n\to \infty}\int_{-\infty}^{+\infty} g_n(x)(h'(x) - xh(x))\ dx = 0$$ I'm asked to show that for every bounded continuous function $f$, we have $$ \lim_{n \to \infty} \int_{-\infty}^{+\infty} g_n(x) f(x)\ dx = \int_{-\infty}^{+\infty} G(x)f(x)\ dx $$ I can only guess for the first step that we have $$ \lim_{n\to \infty}\int_{-\infty}^{+\infty} g_n(x)e^{\frac{x^2}{2}}(e^{-\frac{x^2}{2}}h(x))'\ dx = 0 $$ and I guess it has some connections to probability theory. But I don't know how to continue. Thanks for any help!