For $(X_n)$ i.i.d. standard normal , find $(a_n)$ such that $\frac1{a_n}\max(X_1, ..., X_n)\to1$ in probability

Question

Let $(X_n)$ be independent $N(0,1)$ random variables. Find a numerical sequence $(a_n)$ such that $\frac1{a_n}\max(X_1, ..., X_n)$ converges in probability to $1$ as $n \to \infty$.

I'm not quite sure how to approach this problem. The only thing that comes to mind is some sort of Law of Large Numbers argument, but I'm not sure if that would be useful here.

Answer 1

Hint: Let $M_n=\max\{X_k\,;\,1\leqslant k\leqslant n\}$. Compute $P[M_n\leqslant x]$. Find some sequence $(a_n)$ such that $P[M_n\leqslant a_nx]\to0$ if $x\lt1$ and $P[M_n\leqslant a_nx]\to1$ if $x\gt1$.