Prove that $\lim_{n\to\infty}\Bbb P[\max_{k\le n}x_k \le\sqrt{2\log n-\log(2\log n)-log 2\pi +2x}]=\exp({-e^{-x}})$, where $x_1, x_2$, etc. are independent with common density $(2\pi)^{-1/2}e^{-x^2/2}$ .
I have already proved $\Bbb P[\max_{k\le n}x_k<(2+)\sqrt{\log n}$ as $n\to \infty]=1$ using Borel-Cantelli's lemma. However, it doesn't work here, since the RHS is $\exp({-e^{-x}})$ instead of $1$.
This one seems much more difficult. I really have no clue about this one.
Could anyone kindly provide some help? Thanks!