Let $(X,||\cdot||_X)$ be a normed separable space and let $F$ be a closed subset of $X$, then $F$ is separable.
My attempt
Let $D\subseteq X$ countable and dense, that is $cl_X(D)=X$, then $D\cap F\subseteq F$ is countable. To show that $F$ is separable we must prove that $cl_F(D\cap F)=F$. Now we have
$$cl_F(D\cap F)=cl_X(D\cap F)\cap F\subseteq cl_X(D)\cap cl_X(F)\cap F=X\cap F=F,$$ the we obtained that $cl_F(D\cap F)\subseteq F.$
It remains to prove that $cl_F(D\cap F)\supseteq F.$ We have that $$F=F\cap X=F\cap cl_X(D)\supseteq F\cap cl_X(D\cap F)=cl_F(D\cap F)$$
It's correct?