Showing that there exists $S\in A:|\mu(C\Delta S)|<\epsilon$ and $|\mu(C)-\mu(S)| < \epsilon$ if $A$ is an algebra generating the sigma algebra $F$

Question

Let $(X, F, \mu)$ be a probability space, $A$ an algebra generating $F$, $C\in F$ and $\epsilon > 0$ be given. My probability theory material claims (without a proof) that

As $A$ generates $F$ there exists $S \in A$ s.t. $\mu(C\Delta S) < \epsilon$. Hence $|\mu(C) - \mu(S)| < \epsilon$.

Intuitively this claim makes sense: An algebra is essentially a finite version of a $\sigma$-algebra. Therefore we can approximate, w.r.t. a measure $\mu$, to an arbitrary precision any element of $F$ with elements of $A$. But then the question becomes how one can actually show this without hand-waving.