Let $S$ be a nonempty subset in $\mathbb R^m$ without accumulation points in $\mathbb R^m$. Is then $$ \inf \{ \|x-y\|: x,y \in S, x\neq y \} >0 \textrm{ ? } $$
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Counterexample : $A=\lbrace k; k+\frac{1}{k} | k\geq 1\rbrace$.
Ewan Delanoy
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Thanks. What about the case $m=1$. This counterexample does not work in this case? – user111 Oct 27 '13 at 11:55
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@user111 I’ve rewritten it so that it works in any dimension. – Ewan Delanoy Oct 27 '13 at 11:57