$f$ is a continuous function such that the function $g: \mathbb{R}^+ \to \mathbb{N}$ given by $g(r)$ being the minimal $N$ such that $|f(rn)| < \frac{1}{1989}$ for all $n \ge N$ is well defined. Prove that $T = \{r(g(r)+n) | n \in \mathbb{N}, r > 0\}$ is eventually dense, i.e. there exists $N'$ such that it is dense on $[N', \infty).$
After messing around with quantifiers for a little bit, I showed that another tricky problem was equivalent to this tricky problem. I finished it off as follows:
$\mathbb{R}^+ = \bigcup\limits_n g^{-1}(n).$ Let $S_n = [1,2] \cap g^{-1}(n).$ If all preimages are nowhere dense, then $[1,2] = \bigcup\limits_n S_n$ is a countable union of nowhere dense sets, contradicting the Baire Category Theorem. Thus, there exists $m$ such that $S = g^{-1}(m)$ is dense. But then $\{mr : r \in S\} = mS$ is a dense subset of $T,$ so $T$ is dense.
Can we solve the problem without the Baire Category Theorem? It seems like overkill given that we didn't even use the fact that $f$ is continuous.