Completeness and separability of Lévy's metric 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.
 A: Here is a sketch.
For separability: try functions of the form
$$F(x) = \sum_{i=1}^n a_i 1_{(b_i, \infty)}$$
where $a_i, b_i$ are rational.  (This corresponds to measures which put a rational amount of mass at each of a finite set of rational points.)
For completeness: It suffices to show every Cauchy sequence has a convergent subsequence.  Given a Cauchy sequence $\{F_n\}$, use the compactness of $[0,1]$ and a diagonalization argument to extract a subsequence $\{F_{n_k}\}$ such that $F_{n_k}(x)$ converges for every rational $x$.  (Equivalently, this is the compactness of $[0,1]^{\mathbb{Q}}$.)  Call the limit $G : \mathbb{Q} \to [0,1]$.  $G$ is nondecreasing so it corresponds to a unique nondecreasing, left-continuous $F : \mathbb{R} \to [0,1]$.  (Make this correspondence precise.)  Check that $F \in D$ (this will require using that the sequence is Cauchy).  Now show that $F_{n_k} \to F$ in the $d$ metric.
There is a fair amount of work involved in filling in the details, but I leave it to you.
