The Nielsen–Schreier theorem states (in part):
Let $F$ be a free group, and $H\le F$ be any subgroup. Then $H$ is isomorphic to a free group.
I have seen the topological proof of this theorem using the correspondence between coverings and subgroups of the fundamental group. This has always struck me as using rather strong theory for what it is being used to prove (though I do appreciate the beauty of the argument).
In my head, I see the following (loose) argument:
Let $H$ be a subgroup of $F$, and assume that $H$ is not free. Then there exists some nontrivial relation $h_1h_2\dots h_n = 1$. But then this is also a nontrivial relation in $F$ implying that $F$ is not free, which is absurd. Thus $H$ is free.
Clearly, there must be some problem with this. What are the stumbling blocks here? An issue I see is that the exact notation a relation has always seemed a little vague to me (some reduced word equal to the identity?), but that doesn't seem like it ought to be a large enough problem to invalidate the argument.