Alternative proof that the sum of two convergent series is convergent For my Real Analysis course I'm being asked to prove this basic concept:
If $\sum_{n=1}^\infty |a_n|$ and $\sum_{n=1}^\infty |b_n|$ converge absolutely then prove that $\sum_{n=1}^\infty (a_n+b_n)$ also converges absolutely.
Now this question is very basic. It can be proved by adapting the proof of the sum law for limits to series instead. I've done this but my professors says the answer is wrong. Instead, he wants to me solve this question without using epsilon-proofs. The only hint he provides is that I must consider the rearrangement of each series and apply the triangle inequality to prove convergence. This logic is incomprehensible to me but he refuses to help any further.
Is there another method to prove this question without using limit laws and epsilon-proofs?
 A: Well, we only want to show that
$$
\sum_{n = 1}^\infty \lvert a_n + b_n \rvert < \infty.
$$
By the triangle inequality we have:
$$
\sum_{n = 1}^\infty \lvert a_n + b_n \rvert \leq \sum_{n = 1}^\infty (\lvert a_n \rvert + \lvert b_n \rvert)
$$
Since both
$$
\sum_{n = 1}^\infty \lvert a_n \rvert \quad \text{ and } \quad \sum_{n = 1}^\infty \lvert b_n \rvert
$$
converge, we can rewrite
$$
\sum_{n = 1}^\infty (\lvert a_n \rvert + \lvert b_n \rvert) = \sum_{n = 1}^\infty \lvert a_n \rvert + \sum_{n = 1}^\infty \lvert b_n \rvert < \infty.
$$
So
$$
\sum_{n = 1}^\infty \lvert a_n + b_n \rvert < \infty
$$
according to the direct comparison test.
In general: If we can prove that some series $\displaystyle \sum_{n = 1}^\infty \lvert c_n \rvert$ is bounded, we know (at least in $\mathbb{R}$) that it converges, since the partial sums
$$
S_m: m \mapsto \sum_{n = 1}^m \lvert c_n \rvert
$$
are bounded and increasing. To see the latter just note that ($m \in \mathbb{N})$:
$$
0 \leq \lvert a_{m+1} - b_{m+1} \rvert \implies S_m \leq S_{m} + \lvert a_{m+1} - b_{m+1} \rvert = S_{m+1}
$$
A: *

*The series is bounded above, as $\sum|a_n+b_n|\le\sum |a_n|+\sum|b_n|$.


*The series forms an increasing sequence, as every term $|a_n+b_n|$ is non-negative.
