Jameson Lucero

2022-07-08

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.

Expert

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, $A\subset \left\{q-{a}_{0}:q\in \mathbb{Q}\right\}$. But the latter is countable