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Let $x\in \Delta$, For any $\epsilon>0$, consider $(x-\epsilon,x+\epsilon)$ Using archemedean propery, $\exists N\in \mathbb N: \frac{1}{3^N}<\epsilon$. Let $\frac{M}{3^N}=\max \{\frac{m}{3^N}:\frac{m}{3^N}<x \space \text{and} \space m\in \mathbb N\}$. So $x\in [\frac{M}{3^N},\frac{M+1}{3^N}]\sub...