# Sum of Summations equals Summation of Sum/Infinite Sequence

## Theorem

Let $R: \Z \to \set {\mathrm T, \mathrm F}$ be a propositional function on the set of integers $\Z$.

Let $\displaystyle \sum_{R \paren i} x_i$ denote a summation over $R$.

Let the fiber of truth of $R$ be infinite.

Let $\displaystyle \sum_{R \paren i} b_i$ and $\displaystyle \sum_{R \paren i} c_i$ be convergent.

Then:

$\displaystyle \sum_{R \paren i} \paren {b_i + c_i} = \sum_{R \paren i} b_i + \sum_{R \paren i} c_i$

## Proof 1

Let $b_i =: a_{i 1}$ and $c_i =: a_{i 2}$.

Then:

 $\displaystyle \sum_{R \left({i}\right)} \left({b_i + c_i}\right)$ $=$ $\displaystyle \sum_{R \left({i}\right)} \left({a_{i 1} + a_{i 2} }\right)$ by definition $\displaystyle$ $=$ $\displaystyle \sum_{R \left({i}\right)} \left({\sum_{1 \mathop \le j \mathop \le 2} a_{i j} }\right)$ Definition of Summation $\displaystyle$ $=$ $\displaystyle \sum_{1 \mathop \le j \mathop \le 2} \left({\sum_{R \left({i}\right)} a_{i j} }\right)$ Exchange of Order of Summation: Finite and Infinite Series $\displaystyle$ $=$ $\displaystyle \sum_{R \left({i}\right)} a_{i 1} + \sum_{R \left({i}\right)} a_{i 2}$ Definition of Summation $\displaystyle$ $=$ $\displaystyle \sum_{R \left({i}\right)} b_i + \sum_{R \left({i}\right)} c_i$ by definition

$\blacksquare$

## Proof 2

By definition, $\displaystyle \sum_{R \paren i} b_i$ and $\displaystyle \sum_{R \paren i} c_i$ are sequences in $\R$.

Hence the result as an instance of Sum Rule for Real Sequences.

$\blacksquare$