Let $(X_n)_{n\ge 1}$ be a sequence of i.i.d standard normal r.v's. I am trying to show that $$ \frac{\max_{1\le i \le n} X_i}{\sqrt{2\log n}}$$ converges to 1 a.s.
Using tail bounds, a common problem to prove is $\limsup_n\frac{X_n}{n}=1$ holds with probability 1. Does the result of interest extend this problem easily or is it something else entirely?
