"Darkening of Tristram" paragon portraits on consoles

Let $\alpha=\limsup\limits_{n\rightarrow\infty} \Vert T^n\Vert$.

You know quite a bit more than just that there is a subsequence of $(\Vert T^n\Vert)$ that converges to $\alpha$. From the definition of $\limsup$, you can deduce that $\alpha$ is the largest number that is a subsequential limit point of the sequence $(\Vert T^n\Vert)$.

So, if $\theta>\alpha$, it would follow that $\bigl|\,\{ n : \Vert T^n\Vert>\theta\}\,\bigr|<\infty$ (that is only finitely many terms of $(\Vert T^n\Vert)$ exceed $\theta$). Otherwise, you could find a subsequence that converges to a (possibly infinite) value $c\ge\theta>\alpha$.

Thus, if $\theta>\alpha$, there is an $m$ such that for all $n\ge m$, we have $\Vert T^n\Vert\le\theta$ (of course, this then implies that $\Vert T^n\Vert^{1/n}\le\theta^{1/n}$ for all $n\ge m$).

Note we need $\theta>\alpha$, here. The statement is not true for arbitrary $\theta$. The author chose $\alpha<\theta<1$.

Looking at your input file you can do something like this, using awk:

awk -F"_" '{print $3}' inputfile | uniq > outputfile

or using grep,

grep -o -E '[0-9]+' inputfile | uniq > outputfile

using sed,

sed 's/[^0-9]*//g;/^\s*$/d' inputfile | uniq