Let $(r\_n)\_{n \ge 1}$ be an enumeration of the rationals. Consider the union $A := \cup\_n (r\_n-\frac{1}{n^2},r\_n+\frac{1}{n^2})$. It is unclear a-priori whether $A$ covers the real line, since although the rationals are dense in the reals, the $\frac{1}{n^2}$'s might shrink too fast. However, using measure theory, it is very easy to see $A$ does not cover much: indeed, $m(A) \le \sum\_n m((r\_n-\frac{1}{n^2},r\_n+\frac{1}{n^2})) = \sum\_n \frac{2}{n^2} = \frac{\pi^2}{3}$. 

Since this argument relies much on the convergence of $\sum\_n \frac{1}{n^2}$, I am wondering whether $B := \cup\_n (r\_n-\frac{1}{n},r\_n+\frac{1}{n})$ covers the whole real line, or what portion of it? Does the amount covered depend on the enumeration we choose?