Let $S^1$ be the unit circle and $L^\infty(S^1)$ the space of measurable functions $f:S^1\to\mathbb{C}$ such that $\|f\|_\infty<\infty$. (In fact $L^\infty(S^1)$ consists of equivalence classes of functions, where $f\sim g$ is they are equal almost everywhere.)
How to show that $L^\infty(S^1)$ is not separable?
$L^\infty(S^1)$ can be thought as the space of $2\pi$ periodic functions. I have been able to show that the space of periodic functions (of arbitrary period) is not separable by showing that the collection of functions of the form $$f_s(x):=e^{isx}$$ for $s\in\mathbb{R}$ is uncountable and satisfies $$\|f_s-f_t\|_\infty=2\delta_{st},$$ but I fail to adapt this to the case of $2\pi$ periodic functions. Is there another approach?
Hint: Consider the indicator functions of intervals.