I have a function $x:\mathbb{R}_+\to \mathbb{N}$, Im sure that for small enough inputs I can bound this function but I dont know how to make a rigorous argument.
Let $a\in\mathbb{R}_+$ and define
$$x(\epsilon):=\Big\lceil \frac{a}{\epsilon} \Big\rceil $$
can I find an $\epsilon_0$ such that for all $\epsilon<\epsilon_0$
$$\epsilon x(\epsilon) \leq a~?$$