Sum of Summations equals Summation of Sum/Infinite Sequence/Proof 2

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


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


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


Then:

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


Proof

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

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

$\blacksquare$