A normed space is not separable if and only if it contains an uncountable set of disjoint balls of the same radius

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?

• Could you edit to write out what you mean by "$E$...countable"? I'm not sure I'm understanding it correctly. Also, it would be nice if you'd proofread to check the spelling. – Nate Eldredge Apr 4 '13 at 14:31
• You should probably rewrite this to clarify the role of $r$. For instance, can a uniform value for $r$ be chosen for all the balls? If so, can we arrange matters so that any fixed $r > 0$ is possible for all the balls, or is it the case that the collection of possible values of $r$ can vary with $E$? – Dave L. Renfro Apr 4 '13 at 15:50