I want some tips how to prove that fact:
Let $A\subset\mathbb{R}$ be a measurable and bounded and $l_{1}\left(A\right)>0$. Show that there exist $a, b\in A$ that $a-b$ is irrational number.
Thanks for help!
2026-03-28 07:00:11.1774681211
Measurable, bounded subset of $\mathbb{R}$ and irrational number
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Suppose not. Fix some $a_0 \in A$. Then by our supposition, $a_0 - a$ is rational for every $a \in A$. In other words, $A \subset \{q - a_0 : q \in \mathbb{Q}\}$. But the latter is countable...