Whether $(C[0,1],d_1)$ is separable or not

Question

Consider $C[0,1]$ with integral metric $$d_1(f,g)=\int_0^1 \vert f(x)-g(x) \vert dx$$

Prove or disprove: $(C[0,1],d_1)$ is separable

I know it is separable with the 'sup' metric, since $\Bbb{Q}[x]$ is a countable dense subset. But I don't know this. Can I have a hint?

Answer 1

Hint: $d_1(f,g)\le d_\infty(f,g)$ where $d_\infty$ is the 'sup' metric. If $f_n\stackrel{d_\infty}{\to}f$, then $f_n\stackrel{d_1}{\to}f$.