On Mertens' theorem

Question

I have a problem understanding the proof of Merten's theorem for the case $$\sum_{n=1}^{\infty}a_{n},\sum_{n=1}^{\infty}a_{n} - abs.convergent$$ then $ \sum_{n=1}^{\infty}c_{n}$ is convergent and $$\sum_{n=1}^{\infty}c_{n} = \bigg(\sum_{i=1}^{m}a_{i}\bigg)\bigg(\sum_{j=1}^{l}b_{j}\bigg) $$ and $$\pi : \mathbb{N} \rightarrow\mathbb{N}\times\mathbb{N}, \pi(n) = (\pi_1(n),\pi_2(n))$$ Proof : Let $$I =\sum_{i=1}^{n}|a_{\pi_1(i),\pi_2(i)}|$$ and $m = max({\pi_1(1),\pi_1(2),...,\pi_1(n)})$, $l=max({\pi_2(1),\pi_2(2),...,\pi_2(n)})$ then $$I\le\bigg(\sum_{i=1}^{m}|a_{i}|\bigg)\bigg(\sum_{j=1}^{l}|b_{j}|\bigg)$$ and so on. I don't understand why we come to this inequality. Can somebody help me, please?