Proving that the closed unit sphere of a separable normed space is separable

Question

Let $L$ be a separable normed linear space. Show that its closed unit sphere $S=\{x\in L : \|x\|=1\}$ is also separable.

I am having quite a hard time with this, given a complete space $L$ then I would be able to prove it but how should I go about this now

Edit: Yes I am aware a separable metric space is second countable

Please help!