Spivak's Calculus 2nd ed - Ch. 23 - pg 480 - Thm 9 - Question on the proof The following is a proof for a theorem in Spivak's Calculus   - Ch. 22:

There are two pieces of the proof I don't follow, first is $(1)$ where it states it follows that
$$ \sum_{i \ \text{or}\ j >L}|a_{i}| \cdot |b_{j}| \leq \epsilon$$
comes from. Because I don't see directly how to relate this to the inequality above it in the proof.
And the reasoning for $(3)$. I get that we are given that $\sum_{n=1}^{\infty}a_{n}$ and $\sum_{n=1}^{\infty}b_{n}$ both converge absolutely individually, so are we just using the idea that since they both converge individually then their product will also converge? and we write that using the partial sum notation where since they converge, there exists $i,j > L$ such that
$$\sum_{i=1}^{\infty}a_{i}\sum_{j=1}^{\infty}b_{j} - \sum_{i=1}^{L}a_{i}\sum_{j=1}^{L}b_{j}| < \epsilon$$?
 A: For (1), note that
$$S_{L'}=\left|\sum_{i=1}^{L'}|a_i|\sum_{j=1}^{L'}|b_j| - \sum_{i=1}^{L}|a_i|\sum_{j=1}^{L}|b_j| \right| = \sum_{i=L+1}^{L'}|a_i|\sum_{j=1}^{L'}|b_j|+ \sum_{i=1}^{L}|a_i|\sum_{j=L+1}^{L'}|b_j| < \epsilon$$
Since $S_{L'}$ is a sum of nonnegative terms it is monotone increasing with respect to index $L'$. As it is also bounded above the sequence $(S_{L'})$ converges with
$$\sum_{i \text{ or } j >L}|a_i| |b_j|= \lim_{L' \to \infty}S_{L'} \leqslant \epsilon$$.
A: You have$$\sum_{i\text{ or }j>L}|a_i|.|b_j|\leqslant\frac\varepsilon2$$because, for any finite set $F$ of pairs $(i,j)$ in which $i>L$ or $j>L$, you take $L'$ so large that for every $(i,j)\in F$ you have $i,j\leqslant L'$ (it's possible, since $F$ is finite) and then$$\sum_{(i,j)\in F}|a_i|.|b_j|\leqslant\sum_{i=1}^{L'}|a_i|.\sum_{j=1}^{L'}|b_j|-\sum_{i=1}^L|a_i|.\sum_{j=1}^L|b_j|\leqslant\frac\varepsilon2.$$
The other inequality follws from the fact that$$\lim_{L\to\infty}\sum_{i=1}^L|a_i|.\sum_{j=1}^L|b_j|=\sum_{i=1}^\infty|a_i|.\sum_{j=1}^\infty|b_j|.$$
