Sum of Summations equals Summation of Sum/Infinite Sequence

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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.

$\blacksquare$


Sources

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