This question is a little tricky (for me, at least), since in the textbook the proof of
Theorem 5: Every subgroup of a free group is free.
is not yet provided (even though I've seen such proofs in Magnus et al.'s "Combinatorial Group Theory [...]," in Lyndon & Schupp's "Combinatorial Group Theory," and in Johnson's "Presentation$\color{red}{s}$ of Groups").
Here is the exercise:
Let $\Xi$ be a set of non-commuting variables. Consider the group of units of the ring of formal power series, with integer coefficients, in the non-commuting variables coming from $\Xi$. Prove that
$$F=\operatorname{gp}(1+\xi\mid \xi\in\Xi)$$
is free on $\{1+\xi\mid \xi\in\Xi\}$. (This is a theorem of W. Magnus.)
Before this exercise is the following
Hint:
Before starting out on the proof of Theorem 5, we give some examples of some groups which turn out to be free. The proofs require the criterion (iv), of Theorem 1 of Chapter 1. (Emphasis added.)
Okay, so here is that criterion:
Theorem 1: (iv) Let $G$ be a group and suppose that $X$ generates $G$. If every reduced $X$-word is different from $1$, then $G$ is a free group, freely generated by $X$.
Notation: Let $X\subseteq G$. The subgroup of $G$ generated by $X$ is denoted $\operatorname{gp}(X)$.
This notation is so that such subgroups are not confused with presentations.
Thoughts:
This is a question I think I should be able to do myself, especially since it seems to be a matter of letting $x_\xi:=1+\xi$ - or something to that effect - as there are no restrictions on the elements of $\Xi$. But then I suspect I'd be using Theorem 5, which would result in an annoying anachronism in my understanding of the material in the book.
Please help :)
As I understand it, your job is to apply Theorem 1, not Theorem 5.
So to start, you should take a sequence of elements $\xi_1,...,\xi_n \in \Xi$ and a sequence of exponents $\epsilon_1,...,\epsilon_n \in \{-1,+1\}$, and you should assume that for all $i=1,...,n-1$ if $\xi_i = \xi_{i+1}$ then $\epsilon_i = \epsilon_{i+1}$. Using those assumptions, your job is then to prove that product $$(1 + \xi_1)^{\epsilon_1}\cdots(1 + \xi_n)^{\epsilon_n} \quad (*) $$ is not the identity element of the ring of formal power series. The assumptions were made in order to express that the product $(*)$ is a reduced word in the set of generators $X = \{1 + \xi \mid \xi \in \Xi\}$.
Can you take it from here?