Please if someone with enough reputation could show my image it would be great.
In this limit question, at the point where it states:
Take the limit as $n \rightarrow \infty$ I really don't follow the logic at all. Since $a \in (0,1)$ then as $n \rightarrow \infty$, $|f(x)-f(a^nx)| \rightarrow |f(x)-f(0)|$, did they confuse $f(0)$ with $0$? Because otherwise how do you go from $$|f(x)-f(0)| \leq \frac{\epsilon}{1-a} |x|$$ to $$|f(x)| \leq \frac{\epsilon}{1-a} |x|$$
It is also very strange that the first property was not used.
You should have, for $x$ enough great,
$$|f (x)|=|f (x)-f (0)+f (0)|\le $$ $$|f (x)-f (0)|+|f (0)|\le$$
$$ \frac {\epsilon}{1-a}|x|+|f (0)|$$
thus
$$\frac{|f (x)|}{|x|}\le \frac {\epsilon}{1-a}+\frac {|f (0)|}{|x|} $$
and use the fact that $$\lim_{x\to+\infty}\frac {|f (0)|}{|x|}=0$$
You don't need the first hypothesis.