Generalized Sum with Finite Non-zero Summands

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Theorem

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

Let $\family{g }_{i \in I}$ be an indexed family of elements of $G$.

Let $\set{i \in I : g_i \ne 0_G}$ be finite.

Let $\set{i_1, i_2, \cdots, i_n}$ be a finite enumeration of $\set{i \in I : g_i \ne 0_G}$.

Then the generalized sum $\ds \sum_{i \mathop \in I} g_i$ converges to the summation $\ds \sum_{k \mathop = 1}^n g_{i_k}$

Proof

Let $J = \set{i \in I : g_i \ne 0_G}$.

From Finite Generalized Sum Converges to Summation:

the generalized sum $\ds \sum_{j \mathop \in J} g_j$ converges to the summation $\ds \sum_{k \mathop = 1}^n g_{i_k}$

From Generalized Sum Restricted to Non-zero Summands:

the generalized sum $\ds \sum_{i \mathop \in I} g_i$ converges to the summation $\ds \sum_{k \mathop = 1}^n g_{i_k}$

$\blacksquare$