Claim:
Let $f:[0,\infty) \rightarrow \mathbb{R}$ be a continous function, for which $\text{lim}_{n \rightarrow \infty}f(nt) = 0$ ($n \in \mathbb{N}$) holds for all $t \in [0, \infty)$.
Then $\text{lim}_{t \rightarrow \infty}f(t) = 0$.
I think you need to use Baire's category theorem, so i tried to somehow formulate the condition $\text{lim}_{t \rightarrow \infty}f(t) = 0$ into something with subsets of $[0, \infty)$, and arrived at:
$\text{lim}_{t \rightarrow \infty}f(t) = 0 \iff \forall \epsilon > 0: \ \{f < \epsilon\} \supset [R_{\epsilon},\infty) \text{ for one } R_{\epsilon} \in \mathbb{R}$.
But i don't really have an idea how to proceed.
$ f(x)= \begin{cases} 1&\text{if}, x=\sqrt{2}+n,\ n\in\mathbb{N}\cup{0}\ 0&\text{otherwise} \end{cases} $
– Adam Rubinson Aug 08 '23 at 11:35