$\xi_n \to \xi$. Prove that $\xi$ is normally distributed or a constant.

Question

Let $\{ \xi_n\:|\: n \in \mathbb{N} \} $ be a sequence of normally distributed random variables. Prove, that if $\xi_n \stackrel{d}{\to} \xi$ than $\xi$ is either normally distributed or a constant.

I tried to use that $\varphi_{\xi_n}(t) \to \varphi_{\xi} (t)\:\:\ \forall t \in \mathbb{R}$ where $\varphi_\eta$ defines the characteristic function of $\eta$. However, I did not proceed any further. I will be greatful if you help me.

Answer 1

$$\varphi_{\xi_n}(t) = e^{it\mu_n-\sigma_n^2t^2/2}.$$ If $\varphi_{\xi_n} \to \varphi_\xi$ then $e^{-\sigma^2_n t^2/2} = |\varphi_{\xi_n}(t)| \to |\varphi_\xi(t)|$ for each $t$. So, $\sigma_n^2 \to \sigma^2$ for some $\sigma^2$. Consequently, $\mu_n \to \mu$ for some $\mu$. Then, $\varphi_\xi(t)=e^{it\mu-\sigma^2 t^2/2}$, which is the characteristic function of $N(\mu,\sigma^2)$ if $\sigma^2>0$, or is the characteristic function of the constant $\mu$ if $\sigma^2=0$.