Limit of expectation of maximum of standard normal distribution

Question

I cannot find the way to show that

if $X_1,\dots,X_n$ are independent standard normal random variables, then

$$ \lim_{n \rightarrow \infty} \frac {\mathbb{E} \max_{i=1,\dots,n}X_i}{\sqrt{2\log n}} = 1 $$

I could only show that $\mathbb{E} \max_{i=1,\dots,n}X_i \leq \sqrt{2 \log n}$

Answer 1

See the solved exercise 18.7 (pages 332 and 358) in https://homes.cs.washington.edu/~karlin/GameTheoryBook.pdf