I am reading this interesting explanation (see second proof) of the
Expectation of the maximum of gaussian random variables
And it is stated that $F_{EV}=exp(-exp(-(x-\mu_n)/\sigma_n))$. $$ \mu_n = \Phi^{-1}\left(1-\frac{1}{n} \right) \qquad \qquad \sigma_n = \Phi^{-1}\left(1-\frac{1}{n} \cdot \mathrm{e}^{-1}\right)- \Phi^{-1}\left(1-\frac{1}{n} \right) $$ Here $\Phi^{-1}(q)$ denotes the inverse cdf of the standard normal distribution.
I would like to figure out where $F_{EV}$ converges to.
It seems that it converges to zero, but I am not sure how to show it.
