Let $D$ be the set of all functions $F: \mathbb{R} \rightarrow \mathbb{R}$ which are nondecreasing, left-hand-side continuous and $\lim_{x \rightarrow -\infty} F(x)=0$ and $\lim_{x \rightarrow \infty} F(x)=1$. Let $d$ be a Lévy metric in $D$, that is:
$d(F,G)=\inf \{ e >0: G(x-e)-e \leq F(x) \leq G(x+e)+e\text{ for }x\in \mathbb{R} \}\;.$
How to prove completeness and separability of $(D, d)$ ?
I know that a sequence $(F_n)$ from $D$ is convergent to $F$ from $D$ iff
$\lim_{n\rightarrow \infty} F_n(x)=F(x)$ in each $x \in \mathbb{R}$ in which $F$ is continuous.