Sum Rule for Convergent Generalized Sums

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Theorem

Let $\struct {G, +}$ be a commutative topological semigroup.

Let $\family {g_i}_{i \mathop \in I}$ and $\family {h_i}_{i \mathop \in I}$ be an indexed family of elements in $G$.

Let the generalized sums $\ds \sum_{i \mathop \in I} g_i$ and $\ds \sum_{i \mathop \in I} h_i$ be convergent to the following limits:

$\ds \sum_{i \mathop \in I} g_i = a$

$\ds \sum_{i \mathop \in I} h_i = b$

Then:

the generalized sum $\ds \sum_{i \mathop \in I} \paren{g_i + h_i}$ converges to the limit:

$\ds \sum_{i \mathop \in I} \paren{g_i + h_i} = a + b$

Proof

Consider the set $\FF$ of finite subsets of $I$.

By definition of Definition:Generalized Sum:

$\ds \sum_{i \mathop \in I} g_i$ is the net $\ds \family{\sum_{i \mathop \in F} g_i}_{F \mathop \in \FF}$

and

$\ds \sum_{i \mathop \in I} h_i$ is the net $\ds \family{\sum_{i \mathop \in F} h_i}_{F \mathop \in \FF}$

By definition of convergence:

$\ds \lim_{F \mathop \in \FF} \paren{\sum_{i \mathop \in F} g_i} = a$

and:

$\ds \lim_{F \mathop \in \FF} \paren{\sum_{i \mathop \in F} h_i} = b$

From Sum Rule for Convergent Nets:

$\ds \lim_{F \mathop \in \FF} \paren{\sum_{i \mathop \in F} g_i + \sum_{i \mathop \in F} h_i} = a + b$

We have:

$\forall F \in \FF : \ds \paren{\sum_{i \mathop \in F} g_i} + \paren{\sum_{i \mathop \in F} h_i} = \sum_{i \mathop \in F} \paren{g_i + h_i}$

Hence:

$\ds \lim_{F \mathop \in \FF} \sum_{i \mathop \in F} \paren{g_i + h_i} = a + b$

By definition of generalized sum, $\ds \sum_{i \mathop \in I} \paren{g_i + h_i}$ converges to the limit:

$\ds \sum_{i \mathop \in I} \paren{g_i + h_i} = a + b$

$\blacksquare$