Alright this question is from CarothersReal Analysis.

Let be an enumeration of . For each each n, let be the open interval centered at of radius , and let . Prove that U is a proper, open subset, dense subset of and that is nowhere dense in .

Where I am stuck is try to show that is a proper subset of . My first thought was a proof by contradiction, by assuming that . Then by the Baire Categroy theorem one of the but this didn't seem to go anywhere.

Thanks for any input