I left out lots of other details, too, because it was meant to be an outline for the original commenter, not a proof. In any case,

  \sum_{i=1}^{\infty} \frac{(-1)^i}{i^2}

is not conditionally convergent, so it is not a counterexample to my original statement. Here I take conditionally convergent to mean "the series converges, but it does not converge absolutely." :)

What's more, if Σa_i is conditionally convergent then the sums of both A_+ and A_- diverge. You're right that one has to use this fact in the full proof.