The following two questions come from some steps of my thought about solving an exercise. Thanks for help.
Let $X_1, X_2, ..., X_n, ...$ be i.i.d $N(0,1)$ random variables.
Define $M_n:=\max_{1\leq i \leq n} X_i$, for all $n\geq 1$.
Do we have that, almost surely, $M_n$ is bounded for all $n$?
One more question, do we have that, almost surely, $M_n\neq 0$ for all n?
Thanks.