Let $E$ be a set of positive Lebesgue measure on the real line. Let $1>\epsilon>0$ be given. Show that there exists an interval $I$ such that $m(E\cap I)>\epsilon m(I)$ where $m$ is the Lebesgue measure on the real line.
I tried answering by contradiction method without success. Then tried writing the inequality as $m(I\backslash E)<(1-\epsilon)m(I)$ which also didn't help much either.