$L^∞(R,L, λ)$ Inseparable?

Question

I know that $L^p(R,L, λ)$ with the $L^p$-norm is a separable space for $1 ≤ p < ∞$ Here if a normed space $X$ has a countable dense subset, then X is said to be separable. For example, R is separable with the usual norm. This is an important property but how about for $L^∞(R,L, λ)$? It should be inseparable but I'm not sure how to prove hm~

Answer 1

We can find an uncountable subset $F$ of elements of $L^p$ such that $f,g \in F$ and $f \neq g$ implies $\|f-g\|_\infty \ge 1$.

If then $D$ is a dense subset of $L^p$ then it must intersect all non-empty open balls of $L^p$, but the open balls $B(f, \frac12), f \in F$ are all disjoint by the triangle inequality and the distance property of $F$, and so $D$ contains at least as many elements as there are these balls, so $|F|$ many. As $F$ is uncountable, $L^p$ cannot have a countable dense set.

Try to build such an $F$ from characteristic functions.