complete proof of non-separability of D space

Question

Let $D$ be a set of càdlàg functions on $[0,1]$.

Define $f_\alpha(\cdot) \equiv 1(\cdot \ge \alpha)$ for $\alpha \in [0,1]$, which is obviously in $D$.

Then, if we denote $|| \cdot ||$ as a uniform metric, then $||f_\alpha - f_\beta||=1$ for any $\alpha \neq \beta$.

I learned this implies non-separability of $D$ space, but I failed to complete the proof.

Who can give me any hint or something? Thanks!

Answer 1

If $(D,\|.\|)$ were separable, any subset would be separable, in particular $\{f_{\alpha}\}_{\alpha\in [0,1]}$. But it's discrete and non-numerable, so it's not separable.