Prove or disprove that if $\sum a_n$ and $\sum b_n$ are both divergent, then $\sum (a_n \pm b_n)$ necessarily diverges.
I decided to disprove this by offering the example:
Let $a_n = n$.
Let $b_n = -n$.
Then $\sum (a_n + b_n) = \sum 0 = 0$, which converges.
However, my question says $\pm$, and in the $-$ case, we have $\sum (a_n - b_n) = \sum 2n = \infty$, which diverges. So, does this still count as a valid counter example or not? I partly proved it wrong so I think it's still valid.