Suppose that the goal is to prove $f(x) \leq 0$ and I construct an infinite sequence $a_n(x) > 0$ and $\lim_{n \to \infty} a_n(x) =0$.
If I can prove $f(x) \leq a_n(x)$ by induction, can I arrive at the conclusion that $f(x) \leq 0$?
For example, let $a_n(x) = \frac{1}{2^n}x$, is this true that $\forall n\in \mathbb N, f(x) \leq a_n(x) \Rightarrow f(x) \leq 0$?