If Q is dense in R, why exist a family of open balls around rational points that does not cover R? Q is dense in R. There are several definitions of what this means. However, it is still the case that we can put open balls around each point in Q without covering R by using balls with rapidly shrinking radius so that the sum of the radii is finite. This seems really counterintuitive. Can someone help explain what's going on? It seems that every point in R should be hit because there are arbitrarily close rational points, but we are still able to make the balls shrink fast enough....

Take a real number which is not rational, say $\pi$. For each number $q\in Q$ let $r_q$ be the distance from $q$ to $\pi$ i.e. $|q-\pi|$. Then the open balls $\{B_{r_q}(q)\}_{q\in Q}$  will cover all of $Q$ but will not cover $\pi$ i.e. we have not covered all of R even though Q is dense in R.
