Prove or give a counter example $ \sum_{n=1}^{\infty } a_{2n} \ and \sum_{n=1}^{\infty } a_{2n-1} $ converge than $ \sum_{n=1}^{\infty } a_{n} $ claim: 
if$$ \sum_{n=1}^{\infty } a_{2n} \  and \sum_{n=1}^{\infty } a_{2n-1} $$
both converge than 
$$ \sum_{n=1}^{\infty } a_{n}  $$
converge.
I managed to prove that this claim is true if $a_n \ge0$
but, what about general series ?
I got the same issue with this claim : 
$$ \sum_{n=1}^{\infty } |a_{n}| converge  $$
than 
$$ \sum_{n=1}^{\infty } a_{n}\ ^ 2  $$
converge. again ,  managed to prove that this claim is true if $a_n \ge0$.
Thanks for helping.
 A: Both statements are true.  For the first: note that any finite sum can be rearranged as
$$
\sum_{n=1}^{N} a_{n} = 
\sum_{n=1}^{\lfloor N/2 \rfloor } a_{2n} + \sum_{n=1}^{\lfloor (N + 1)/2\rfloor} a_{2n-1}
$$
For the second: note that $|a_n| \to 0$, and that $a_n^2 < |a_n|$ whenever $|a_n| < 1$.
A: Using the cauchy criterion...
Fix $\varepsilon > 0$.
If $\sum a_{2n}$ converges, there is a $N_e$ for which $|\sum_{n = p}^{q} a_{2n}| < \varepsilon/2$ for any $q \geq p \geq N_e$.
If $\sum a_{2n-1}$ converges, there is a $N_o$ for which $|\sum_{n = p}^{q} a_{2n-1}| < \varepsilon/2$ for any $q \geq p \geq N_o$.
Now let $N = 2 \max(N_e, N_o)$. For any $q \geq p \geq N$ we have $$|\sum_{n=p}^q a_n| \leq |\sum \limits_{n \text{ odd}, p \leq n \leq q} a_n|  + |\sum_{n \text{ even}, p \leq n \leq q} a_n| < \varepsilon$$
so the main series is convergent as well.

For the second, we know $a_n \to 0$ by convergence, hence past some finite number of terms we have $a_n^2 < |a_n| < 1$, so it converges by comparison.
