Finite Generalized Sum Converges to Summation

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

Let $\set{i_0, i_1, \ldots, i_n}$ be a finite enumeration of a finite set $I$.

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

Then:
 * the generalized sum $\ds \sum_{i \in I} g_i$

converges to:
 * the summation $\ds \sum_{k =0}^n g_{i_k}$