Free monoids, length of a word $w$

Question

Let $A$ be a finite alphabet with $|A|=a$, let $S$ be a (not neccesarily finite) set of non-empty words in the free monoid $A^*$ such that $S$ generates a free monoid of $A^*$. Prove that $$\sum\limits_{w\in S}\frac{1}{a^{l(w)}}\leq 1$$ where $l(w)$ is the length of word $w$.