# The distribution of the maximum of a series of extreme value type I random variable

I have an infinite series of independent identically distributed random variables $\{X_i\}_{i=1}^\infty$ which follows extreme value type I distribution which can be found [here] (https://en.wikipedia.org/wiki/Gumbel_distribution), then I was wondering what is the distribution of $Y:=\underset{i\geq 1}{\max}~X_i$?

For any $n$, $$\Pr(Y\le n)=\Pr(X_i\le n\,(\forall i))=\prod_i\Pr(X_i\le n)=\lim_{i\to\infty}\Pr(X_1\le n)^i=0.$$ So $Y=\infty$ with probability 1.