Convergence of series by adding parenthesis Given a series $\sum_{n=1}^{\infty}a_n$
If we know that by inserting parenthesis to the sum, while in each pair of parenthesis all elements are of the same sign we get a convergent series, then the original series is also convergent.
My lecturer proved this theorem in a very complex way I could not follow, does anyone know where I can find a proof online?
of course I'll be happy if someone wants to prove it here but I think it's a pretty long proof.

Let $s_n=\sum_{k=1}^na_n$ and let $S_n$ be the $n$th partial sum of the series that you get after adding the parenthesis. Suppose, say the the first four $a_k$'s are postive, that the five next ones are negative, and then there are some more positive terms. Then $S_1=s_4$. And, after that, $S_2=s_9$. Besides, $s_1\leqslant s_2\leqslant s_3\leqslant s_4=S_1$. And then $s_4\geqslant s_5\geqslant s_6\geqslant s_7\geqslant s_8\geqslant s_9=S_2$. And so on. Each $s_k$ is between a $S_l$ and $S_{l+1}$. So, since $(S_n)_{n\in\Bbb N}$ converges, so does $(s_n)_{n\in\Bbb N}$.
