Generalized Sum of Constant Zero Converges to Zero

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 $0_G$