How to prove this asymptotic behavior of the expectation of the maximum of the independent standard normal variables?

Question

How to prove this famous theorem?
If $G_1,G_2,...,G_n$ are independent standard normal random variables($\mathcal{N}(0,1)$), then: $$\lim_{n\rightarrow\infty}\frac{\mathbb{E}[\max_{i=1,2,...,n}G_i]}{\sqrt{2\log n}}=1$$ I know the first step:
$\Psi_G(\lambda)=\mathbb{E}[e^{\lambda G}]=\frac{\lambda^2}{2}$ then according to the Jensen's inequality, for all $\lambda>0$ \begin{equation*} \begin{split} \exp(\lambda\mathbb{E}[\max_{i=1,2,...,n}G_i]&\leq\mathbb{E}[\exp(\lambda\max_{i=1,2,...,n}G_i)]\\ &=\mathbb{E}[\max_{i=1,2,...,n}\exp(\lambda G_i)]\\ &\leq \sum_{i=1}^n\mathbb{E}[\exp(\lambda G_i)]\leq ne^{\frac{\lambda^2}{2}} \end{split} \end{equation*} then $$\mathbb{E}[\max_{i=1,2,...,n}G_i]\leq\frac{\log n}{\lambda}+\frac{\lambda}{2}$$ the right term is minimised with $\lambda=\sqrt{2\log n}$ we have $$\mathbb{E}[\max_{i=1,2,...,n}G_i]\leq\sqrt{2\log n}$$ What's the next step?