A set constructed with the rational numbers is dense in a certain set.

Question

Let, $$\mathbb{Q} = \{q_1,\dots,q_n,\dots \}.$$

Let, $$I_{n} = q_n + (-2^{-n},2^{-n}).$$

Let, $B \subseteq \mathbb{R}$ such that $\mu(B^{c})=0$. Then we define $$U := \bigcup_{n=1}^{\infty}I_n.$$

Is $(U \cap B)$ dense in $B$? I'm having trouble showing this one to be true.

Thanks!