Separable Bnach space implies that the unit ball is separable

Question

if someone could give me an idea of ​​how to carry out this demonstration I would greatly appreciate it, let E be a Banach space, demonstrate

E is separables $\Rightarrow$ The closed unit ball $B_E=\{x∈E:||x||≤1\}$ is separables

Answer 1

Let $A$ be a countable dense subset of $E$. Then for every positive rational $r\le 1$, $A_r'=\{\frac{ra}{\|a\|}\mid a\in A\}$ is countable and dense on the sphere of radius $r$. Then the union $\cup_r A_r'$ is a countable dense subset of the unit ball.