 Logarithmic-tail Fréchet normalization

For a distribution function with tail $1-F(x)=1/(x\log x)$, let $a_n>1$ solve $a_n\log a_n=n$. Then $n(1-F(a_{n}x))\to 1/x$ for $x>0$, so the normalized sample maximum tends to the shape-one Fréchet distribution. The inequalities $n/\log n<a_n<n/(\log n-\log \log n)$ for large $n$ show $a_n\sim n/\log n$, allowing that simpler scale too.