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So this problem is in line with the post of this week, Baire’s Lemma. Have a go and let me know your thoughts in the comments!

Let $D\subset (0,\infty)$ be unbounded and open. Then let the set $$G=\{x\in (0,\infty); nx\in D ~\text{for infinitely many}~n\in\mathbb{N}\}.$$Prove that $G$ is dense in $(0,\infty)$.