If a series is conditionally convergent, then the series of positive and negative terms are both divergent

Question

Let $ \sum_{n=1}^\infty a_n $ be a conditionally convergent series. Then the series $\sum_{n=1}^\infty b_n$ of positive terms of $\{a_n\}_{n=1}^\infty$ and $\sum_{n=1}^\infty c_n$ of negative terms are divergent.

My proof: if $\sum_{n=1}^\infty a_n$ is conditionally convergent, then $\sum_{n=1}^\infty |a_n| = \infty$. Let's suppose that $\sum_{n=1}^\infty b_n$ and $\sum_{n=1}^\infty c_n$ are convergent, so since they are composed of positive and negative terms respectively, then $\sum_{n=1}^\infty |b_n| < \infty$ and $\sum_{n=1}^\infty |c_n| <\infty$ and we have that $$ \sum_{n=1}^\infty |a_n| = \sum_{n=1}^\infty |b_n| + \sum_{n=1}^\infty |c_n| < \infty $$ which is absurd since we supposed that the series was conditionally convergent.

Edit

Can I define $b_n := \left\{\begin{array}{ll} a_n &\text{if } a_n > 0 \\ 0 &\text{otherwise} \end{array} \right.$ and $c_n$ analogously for negative terms and say that $\sum a_n = \sum b_n + \sum c_n $ and, since $\sum a_n$ converges then both $\sum b_n, \sum c_n$ converge? Are the new series equal to the series of positive (and negative) terms?

Is this correct? Thanks in advance

Answer 1

Note that $$b_n=\dfrac{a_n+|a_n|}{2} \;\;(\geqslant 0), \\ c_n=\dfrac{a_n-|a_n|}{2} \;\;(\leqslant 0) $$ and $$c_n=a_n-b_n.$$ If $\sum a_n$ converges (conditionally) and $\sum b_n$ is convergent (absolutely) then $\sum {c_n}$ is convergent (absolutely). Because $a_n=b_n+c_n$ then $\sum a_n$ must be absolutely convergent, which contradicts to its conditional convergence.

Answer 2

As Jonas Meyer said, you haven't proved what you set out to prove, that neither the $b$ series nor the $c$ series converges. What you have proved, though, is that they can't both converge. So you could complete the proof if you could somehow exclude the possibility that one of the two converges and the other diverges.

Answer 3

This is incorrect, because the negation of [$P$ and $Q$] is [(not $P$) or (not $Q$)]. So if you are going to argue by contradiction, you would suppose that $\sum b_n$ converges OR $\sum c_n$ converges.