Can someone explain the last sentence to me?
It says we get a contradiction if only one of the sums have an unbounded sum. I don't see an issue.
$$\sum a_n = \sum (p_n + q_n)$$
$$\sum |a_n| = \sum (p_n - q_n)$$
If both of the sums $\sum p_n$ and $\sum q_n$ diverge, this still makes the whole sum $\sum a_n$ diverge, yielding another contradiction.