Prove that in $R^n$ with the euclidean norm there are no two identical open balls $B_r(q)$ and $B_r(p)$ such that $p\ne q$.
I believe to do this I need to show that if you have balls around two different points of the same radii, that this implies there is some point in one but not the other. However, I can't come up with how to come up with this point in the general case.