1. ## Baire Category Theorem?

Alright this question is from Carothers Real 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

2. Hello,

I don't know if it leads anywhere, but you could see if $\displaystyle U$ is a closed set or not.
If it's not, then it cannot be $\displaystyle \mathbb{R}$ nor $\displaystyle \emptyset$, which are both open and closed.

3. The total length of the intervals in U is 2, so it's a very long way from being the whole of $\displaystyle \mathbb{R}$.

More precisely, the Lebesgue measure of U is at most 2, while the measure of $\displaystyle \mathbb{R}$ is infinite.