Sum of Summations equals Summation of Sum/Infinite Sequence/Proof 2
Jump to navigation
Jump to search
Theorem
Let $R: \Z \to \set {\T, \F}$ be a propositional function on the set of integers $\Z$.
Let $\ds \sum_{\map R i} x_i$ denote a summation over $R$.
Let the fiber of truth of $R$ be infinite.
Let $\ds \sum_{\map R i} b_i$ and $\ds \sum_{\map R i} c_i$ be convergent.
Then:
- $\ds \sum_{\map R i} \paren {b_i + c_i} = \sum_{\map R i} b_i + \sum_{\map R i} c_i$
Proof
By definition, $\ds \sum_{\map R i} b_i$ and $\ds \sum_{\map R i} c_i$ are sequences in $\R$.
Hence the result as an instance of Sum Rule for Real Sequences.
$\blacksquare$