I want to show normed space $E$ is not separable iff $E$ contains uncountable set of pairwise non-intersecting balls of radius $r > 0$.
Use contraposive: first prove $E$ is separable then $E...$countable There is uncoutablely many elments in $E$, removing any elements in the uncoutable set will end up in a subset whose closure is not $E$. Contradict $E$ is dense.
The other direction: $E...$countable then $E$ is separable. How to prove this direction?
Any comments about the first proof? Many thanks!