Let $\mu$ be the Lebesgue measure on $\Bbb{R}$ and $A$ be a measurable subset of $\Bbb{R}$ with positive measure. Prove for any $c\in (0,1),$ $\exists x,y \in A$ st $\mu(A \cap (x,y))>c(x-y)$.
I saw the similar problem where it had slight modification, it says Let $\mu$ be the Lebesgue measure on $\Bbb{R}$ and $A$ be a measurable subset of $\Bbb{R}$ with positive measure. Prove for any $c\in (0,1),\exists$ an interval $I$ st $\mu(A \cap I)>c \mu(I)$. I'm not able to modify the proof since here $I$ may have endpoints which are not in $A$.