Let $\displaystyle F$ be a free group (not necessarily finitely generated). I am trying to prove that $\displaystyle [F, F]$ has infinite index, and I have, so far, shown that this is equivalent to proving that $\displaystyle F^{\prime} = F/[F, F] = <x_1, \ldots, x_n, \ldots : [x_i, x_j]>$ contains no torsion. Which is obvious.

Except...I can't seem to prove it...