Consider $Y_1, \dots, Y_n$ random variables with $n^{1/2}(\bar{Y}_{n} - \mu ) \xrightarrow{D} \mathcal{N}(0,\,\sigma^{2})$
It holds that $g_{n}(x_n(c))=\Phi(c-n^{1/2}\frac{(\bar{Y}_{n} - \mu )}{\sigma}) \xrightarrow{D} \Phi(c-Z), \quad Z \sim \mathcal{N}(0,1)$ where $\Phi$ is cdf of standard normal
In (1) and (2), is the first equality correct and if yes why is that the case and what is the underlying result?
(1): $\lim_{n \rightarrow \infty}E[g_{n}(x_{n}(c))]=E[\Phi(c-Z)]=\int_{\mathbb{R}}\Phi(c-z)\phi(z)dz$
(2): $\lim_{n \rightarrow \infty}\operatorname{Var}[g_{n}(x_{n}(c))]=E[\Phi(c-Z)^2]-E[\Phi(c-Z)]^2=\int_{\mathbb{R}}\Phi(c-z)\phi(z)dz-[\int_{\mathbb{R}}\Phi(c-z)\phi(z)dz]^2$