Why is it that $S \subset (sa^n: s \in S, n > 0)$ for any $a \in A$?

Question

Assuming A is a commutative ring with identity and S is a maximal multiplicative closed subset of A, why is it that $S \subset (sa^n: s \in S, n > 0)$ for any $a \in A$?

This is part of a bigger problem that follows quite easily from this step.