Exercise 34, § 7

(a) Let $(H_i)$ be a decreasing filtered family of distinguished subgroups of a group

G. Suppose that $\bigcap_i H_i = \{e\}$. Now prove 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. Now prove that, if all the $G_\alpha$ are residually finite (cf. I, p. 129, exercise 5), the same is true of $G$. (First reduce to the case where the family is finite, then to the case where it has two elements; thanks to a), one may suppose that the $G_\alpha$ are finite, and one applies the preceding exercise.) (c) Deduce that every free group is residually finite.