Separable Banach space, unit ball, unit sphere demonstration

Question

hello I have the following question, about a Banach space E, prove they are equivalent:

1.E is separables

2.The closed unit ball $B_{E}=\{x\in E :||x||\leq 1 \}$ is separables

3. The closed unit ball $S_{E}=\{x\in E :||x||= 1 \}$ is separables.

I already demonstrated 1) $\Longrightarrow$ 2) and 3) $\Longrightarrow$ 1) but I have no idea how to show 2)$\Longrightarrow$ 3) if anyone has any idea how to demonstrate it, I would greatly appreciate it.

Answer 1

Let $(x_n)$ be a countable dense set in $B_E$. Omitting $0$ still makes it dense, so we may suppose $x_n \neq 0$ for all $n$. Consider $(y_n)$ where $y_n =\frac {x_n} {\|x_n\|}$. If $\|x\|=1$ and $\epsilon >0$ there exists $n$ such that $\|x-x_n\| <\epsilon$. Now $\|x-y_n\| <\epsilon +\|x_n-y_n\|$ and $\|x_n-y_n\|=\|x_n\| \left|{1-\frac 1 {\|x_n\|}}\right|= {1-\|x_n\|}$. Finally note that $\|x_n\| >\|x\|-\epsilon=1-\epsilon$.