I'm trying to follow an argument in C. Giraud's "**High Dimensional Statistics**" (2nd Ed, p. 11 / $\S$ 1.2.3). The specific page is accessible via Google Books [here](https://www.google.com/books/edition/Introduction_to_High_Dimensional_Statist/YiM_EAAAQBAJ?hl=en&gbpv=1&dq=high%20dimensional%20statistics&pg=PR19&printsec=frontcover) but the formatting is awful.


Suppose that $\epsilon\_j \sim {\cal N}(0,1), j = 1, \ldots, p$ and these are all independent.


Then $\mathbb{P}[ \max\_j |\epsilon\_j| \geq x] = 1 - (1 - \mathbb{P}[|\epsilon\_1| > x])^p$. I understand the steps to get the expression on the right from the one on the left.


What I don't understand is the next part, where a limit is taken (?) in $p$:


$\mathbb{P}[ \max\_j |\epsilon\_j| \geq x] = 1 - (1 - \mathbb{P}[|\epsilon\_1| \geq x])^p \sim^{p \rightarrow \infty } p \mathbb{P}[ |\epsilon\_1| \geq x]$


I guess the result is approximate but I cannot grasp the necessary steps. Any help is appreciated!



tail probability of max of Gaussians
The correct version of this formula is
$$P(\max\_j |\epsilon\_j| \ge x) = 1-(1-P(|\epsilon\_1| \ge x))^p 
\underset{x \to\infty}\sim p \,P(|\epsilon\_1| \ge x)$$
for each real $p>0$,
which follows because $(1-u)^p=1-(p+o(1))u$ as $u\to0$.


(The reproduction quality of the preview of the book is indeed terrible. It is also clear that $1-(1-P(|\epsilon\_1| \ge x))^p 
\underset{p\to\infty}\to1$ for each real $x$.)


