Open cover rationals proper subset of R? If I were to cover each rational number by a non-empty open interval, would their union always be R? It seems correct to me intuitively, but I am quite certain it is wrong. Thanks

Take an enumeration of rational let's say $a_m$ and then cover the rationals like this 
$A_m=(a_m-\frac{1}{m^2},a_m+\frac{1}{m^2})$
Now since $ \sum \frac{1}{m^2}$ converges you cannot cover the real numbers
