Df $F$ with regularly varying Tail $\overline{F}$ with index $-\alpha$ for some $\alpha>0$ is in the MDA of the Frechet Distribution

I am reading the proof of Theorem 3.3.7 in the book "Modelling Extremal Events" by Embrechts, Klüppelberg and Mikosch. I don't understand the first couple of lines of the proof: If $F$ is a df with tail $\overline{F}\in\mathcal{R}_{-\alpha}$ for some $\alpha>0$, i.e. $\lim\limits_{x\to\infty}\frac{\overline{F}(tx)}{\overline{F}(x)}=t^{-\alpha}$ for all $t>0$, then how do you show that $\lim\limits_{n\to\infty}n\overline{F}(c_n)=1$, where $c_n:=F^{-1}\left(1-\frac{1}{n}\right)$ and $F^{-1}$ is the generalized distribution function of $F$, defined by $F^{-1}(y):=\inf\left\{x\in\mathbb{R}:F(x)\ge1-\frac{1}{n}\right\}$. Also how does $\lim\limits_{n\to\infty}\overline{F}(c_n)=0$ imply $\lim\limits_{n\to\infty}c_n=\infty$? Maybe there is missing the condition $x_F:=\sup\{x\in\mathbb{R}:F(x)<1\}=\infty$. If not, does $\overline{F}\in\mathcal{R}_{-\alpha}$ imply $x_F$?