Let $X$ be a normed space, i want to show the equivalence of
(i) $X$ is separable
(ii) $B_1(0) = \{ x \in X : \|x\| < 1 \}$ is separable
(iii) $K_1(0) = \{ x \in X : \|x\| \le 1 \}$ is separable
(iv) $S_1(0) = \{ x \in X : \|x\| = 1 \}$ is separable
(v) there exists a countable set $A \subseteq X$ with $X = \overline{\operatorname{span}(A)}$
For every separable space, every subset is also separable, so I got (i) => (ii), (i) => (iii), (i) => (iv), (iii) => (iv) and (iii) => (ii). Furthermore I was able to prove (iv) => (v) and (v) <=> (i). But then there is still something left, for example (ii) <=> (iv). Can you please give me hints what would be the best way to prove (i) <=> (ii) <=> (iii) <=> (iv) <=> (v)?