Exercise 34, § 7

(a) Let $(H_i)$ be a right directed family of normal subgroups of a group $G$. Suppose that $\bigcap_i H_i = \{e\}$. Show that, for every group $G'$, the intersection of the kernels of the canonical homomorphisms $G' * G \to G' * (G/H_i)$ is equal to $\{e\}$.

(b) Let $G$ be the free product of a family $(G_\alpha)$ of groups. Show that, if all the $G_\alpha$ are residually finite (cf. § 5, Exercise 5), so is $G$. (Reduce it first to the case where the family is finite, then to the case where it has two elements; by (a) it may be assumed that the $G_\alpha$ are finite, then apply the preceding exercise.)

(c) Deduce that every free group is residually finite.