Is it possible to have a discrete set of real numbers $S$, such that for every $a\in S,\exists$ an infinite sequence $\{a_i\}_{i\ge 1}$ such that $$\lim_{n\to\infty}a_n=a$$
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1Take perfect sets. Then all points in a perfect set are limit points. Example Cantor set. – Rick Jul 21 '19 at 14:27
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Yes, The set of rational numbers is not an interval and every rational number $r$ is the limit of the sequence $r+1/n$
Mohammad Riazi-Kermani
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