Generalized Sum of Constant Zero Converges to Zero

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Theorem

Let $G$ be a commutative topological semigroup with identity $0_G$.

Let $\family{g_i}_{i \in I}$ be the indexed family of $G$ defined by:

$\forall i \in I : g_i = 0_G$

Then:

the generalized sum $\ds \sum_{i \mathop \in I} g_i$ converges to $0_G$

Proof

Let $\FF$ denote the set of finite subsets of $I$.

From Power of Identity is Identity:

$\forall F \in \FF : \ds \map \phi F = \sum_{i \mathop \in F} g_i = 0_G$

Hence the net $\ds \sum \set {g_i: i \in I}$ is a constant mapping.

From Constant Net is Convergent:

the generalized sum $\ds \sum_{i \mathop \in I} g_i$ converges to $0_G$

$\blacksquare$