Absolute convergence theorem says that if $\sum |a_n|$ is convergent, so is $\sum a_n$
We have by triangle inequality :
$$|a_1+a_2+\cdots+a_n|\le |a_1|+|a_2|+\cdots+|a_n|$$
Now if $S_n$ is the sequence of partial sums of $\sum a_n$ and $P_n$ is sequence of partial sums of $\sum|a_n|$ and assuming $P_{\infty}=L$, we have:
$$|S_n|\le P_n$$
Taking limit we get:
$$\left|\lim_{n\to \infty}S_n\right|\le L$$
Hence:
$$-L \le S_{\infty}\le L$$
Hence $\sum a_n$ is convergent.
On
We can use that
$$b_n=-|a_n|\le a_n \le |a_n|=c_n$$
and conclude that
$$\sum |a_n|<\infty \implies \sum a_n<\infty$$
indeed $\sum (c_n - b_n)$ converges and therefore since
$$0\le a_n-b_n \le c_n-b_n$$
also $\sum (a_n - b_n)$ converges and since
$$\sum a_n = \sum(b_n+(a_n-b_n))= \sum b_n+\sum (a_n-b_n)$$
we conclude that $\sum a_n$ is the sum of two convergent series.
Hint:
Show instead that the Cauchy criterion is satisfied using
$$|S_m-S_n|=\left|\sum_{k=n+1}^m a_k\right| \leqslant \sum_{k=n+1}^m |a_k|$$