Convergent Series can be Added Term by Term

Theorem
Let $\ds \sum_{n \mathop = 1}^\infty a_n$ and $\ds \sum_{n \mathop = 1}^\infty b_n$ be two convergent series converging to $A$ and $B$ respectively.

Then:
 * $\ds \sum_{n \mathop = 1}^\infty \paren {a_n + b_n} = A + B$