Limit of the set $(f(ax),f(x)]$ as $x\to\infty$ and $f$ non-increasing

Limit of the set $(f(ax),f(x)]$ as $x\to\infty$ and $f$ non-increasing

Assume $f>0$ is non-increasing with $\lim_{x\to\infty}f(x)=0$ and
$a\in(1,\infty)$. I assume that $$\limsup_{x\to\infty} (f(ax),f(x)]
=\emptyset.$$ How can I prove it?
I know that I have to show that for any sequence $x_n\to\infty$ $$
\bigcap_{n=1}^\infty\bigcup_{m\geq n} (f(ax_m),f(x_m)] = \emptyset, $$ but
I don't see how to do it. Any help would be appreciated.
I think I could show that the indicator functions of such sets converge to
zero by showing that for any $y>0$ I can find $n\in\mathbb{N}$ large
enough such that $f(x_n)< y$, implying there is no element in the limit
set (since $0\notin (f(ax_n),f(x_n)]$ for all $n$).
Edit: I also know, that the second approach is equivalent to the first,
still, I would like to see how one can prove it using sets.