Alright this question is from CarothersReal Analysis.

Let $\displaystyle (r_n)$ be an enumeration of $\displaystyle \mathbb{Q}$. For each each n, let $\displaystyle I_n$ be the open interval centered at $\displaystyle r_n$ of radius $\displaystyle 2^{-n}$, and let $\displaystyle U= \cup_{n=1}^{\infty}I_n$. Prove that U is a proper, open subset, dense subset of $\displaystyle \mathbb{R}$ and that $\displaystyle U^{c}$ is nowhere dense in $\displaystyle \mathbb{R}$.

Where I am stuck is try to show that $\displaystyle U$ is a proper subset of $\displaystyle \mathbb{R}$. My first thought was a proof by contradiction, by assuming that $\displaystyle \cup_{n=1}^{\infty}I_n=\mathbb{R}$. Then by the Baire Categroy theorem one of the $\displaystyle int\{ \overline{I}_n\} \ne \emptyset $ but this didn't seem to go anywhere.

Thanks for any input