Let $Q_1 \subset \mathbb{R}^d$ be the d-dimensional unit cube ($Q_1 = [0,1]^d$)
Suppose that $E \subset Q_1$ and that $m(E) = 1$. Prove that $E$ is dense in $Q_1$.
I know that $m(Q_1) = 1 = m(E)$. And I know that I have to show that every neighborhood around an $x \in Q_1$ contains a point of $E$. However, I am not quite sure how to go about this. Can anyone help me out here?