Manipulation of Absolutely Convergent Series

Theorem
If a sum is absolutely convergent, all of the following are true:

$$\sum_{n=1}^\infty a_n = \sum_{n\in \N} a_n \ $$, where in the second expression the terms are taken in any order;

$$\sum_{n=1}^\infty a_n \chi_A = \sum_{n\in A} a_n \ $$, where $$A \ $$ is a set and $$\chi_A \ $$ its characteristic function;

$$b \sum a_n = \sum ba_n \ $$.