Let $S$ be a finite set of real numbers. For any $\varepsilon > 0,$ is it always possible to choose $\alpha > 0$ such that every element of $S$ multiplied by $\alpha$ is withing $\varepsilon$ of a nonnegative integer?
2026-03-27 22:57:33.1774652253
Can we scale the elements of a set of real numbers so that they are arbitrarily close to integers?
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Yes, it is always possible. Let $m$ be an element of $S.$ We can make an arithmetic progression with difference $1/m,$ and if $\alpha$ is within $\varepsilon/m$ of a nonzero term, then the scaled version of $m$ would be within $\varepsilon$ of an integer. Let $n$ be the maximum element of $S$. If $\alpha$ is within $\varepsilon/n$ of a nonzero term of every such constructed arithmetic sequence, then we are done. Such a value of $\alpha$ can be found by the answer to this question.