= Solution
For large $t$, write the <survival function> as
$$
\overline F(t)=t^{-2}\ell(t),\qquad \ell(t)=\frac{C}{\log t\,\log\log t}.
$$
A positive <function> $\ell$ is <slowly varying> if $\ell(tu)/\ell(t)\to1$ for every fixed $u>0$. Here
$$
\frac{\ell(tu)}{\ell(t)}
=\frac{\log t\,\log\log t}{\log(tu)\,\log\log(tu)}\longrightarrow1,
$$
since $\log(tu)=\log t+\log u$ and $\log\log(tu)-\log\log t\to0$. Thus $\overline F$ has <regular variation> of index $-2$.
The relevant extreme-value criterion is that an infinite right endpoint and a <survival function> regularly varying with index $-r$, $r>0$, give attraction to the standard <Fréchet distribution> of shape $r$. Its <distribution function> is $G_r(x)=e^{-x^{-r}}$ for $x>0$ and zero for $x\leq0$. Here is a direct verification, so no additional theorem is needed to establish the criterion in this example. Choose $a_n\to\infty$ satisfying $n\overline F(a_n)\to1$ and set $b_n=0$. For $x>0$,
$$
n\overline F(a_nx)
=n\overline F(a_n)\frac{\overline F(a_nx)}{\overline F(a_n)}\longrightarrow x^{-2}.
$$
If $u_n\to0$ and $nu_n\to c$, then $n\log(1-u_n)=-nu_n+O(nu_n^2)\to-c$. Applying this with $u_n=\overline F(a_nx)$ gives $F(a_nx)^n\to e^{-x^{-2}}$. For $x\leq0$, the <distribution function> is zero because the observations are supported on $[10,\infty)$. Therefore
$$
\boxed{G(x)=\begin{cases}e^{-x^{-2}},&x>0,\\0,&x\leq0,\end{cases}\qquad F\in D(G).}
$$
For example, elementary asymptotic normalizers are
$$
a_n=\left(\frac{2Cn}{\log n\,\log\log n}\right)^{1/2},\qquad b_n=0,
$$
for sufficiently large $n$, with arbitrary positive definitions at smaller indices. Indeed $\log a_n\sim\tfrac12\log n$ and $\log\log a_n\sim\log\log n$, so $a_n^2\log a_n\log\log a_n\sim Cn$. The precise valid choice of $C$ at the lower endpoint does not change the limiting shape.
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