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Show that there exist two disjoint closed sets $A, B \subset \mathbb{R}$ such that there does not exist a positive $ϵ$ with $d(x, y)\ge ϵ$ where $x∈A$ and $y∈B$.

I have already proved that if $A$ is compact and $B$ is closed, then there exist a positive $ϵ$ such that $d(x, y)\geϵ$. So for the question that we need to prove, we need to assume that none of $A$ and $B$ are bounded. But how does the unbounded property help to prove this one?

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Let $$A=\{2,3,4\ldots\}\ ,\quad B=\{2\tfrac12,3\tfrac13,4\tfrac14,\ldots\}\ .$$

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  • $\begingroup$ Nice example +1! $\endgroup$ – Vim Feb 5 '16 at 13:49
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$A=\bigcup_{i=2}^\infty \{i\}$, $B=\bigcup_{i=2}^\infty [i+1/i,i+2/3]$, since $i+1/i$ approaches $i$ as closely as we wish for large $i$.

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