Hmm, have you learned the concept of Lebesgue density? There's a theorem that states if E is measurable , then for a.e. x in E

d(x, E) = = 1 almost everywhere where B is a ball that contains x.

Fix an alpha, and suppose that no such interval, I, exists.

Thus, for all x in E, we have:

, which obviously contradicts the above statement.

EDIT: Oops, you never said E was measurable, but I think it would still work with outer measures, let me confirm.