Let a continuous function $f: \mathbb R \to \mathbb R$ such that:$\forall \theta >0,n \in \mathbb N^+$,we have $$\lim_{n\to +\infty}f(n \theta)=0.$$ Prove that $$\lim_{x\to +\infty}f(x)=0$$
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could you update the question with definition of both limits and try to derive one from the other and let's see where you get stuck? – gt6989b Oct 27 '19 at 01:08

1HINT: Baire Category theorem – Marios Gretsas Oct 27 '19 at 01:21

This question has been already answered here: https://math.stackexchange.com/questions/101086/limntoinftyfnx0implieslimxtoinftyfx0 – Fede1 Oct 30 '19 at 11:56