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

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

converges to:
 * the summation over finite index $\ds \sum_{i \mathop \in I} g_i$

Proof
Let $\FF$ be the set of finite subsets of $I$.

Let $h = \ds \sum_{i \mathop \in I} g_i$ be the summation over finite index $I$.

Let $U$ be an open subset of $G$ such that $h \in U$.

From Set is Subset of Itself:
 * $I \in \FF$

Let:
 * $J \in \FF : I \subseteq J$

Let $h'= \ds \sum_{j \mathop \in J} g_j$ be the summation over finite index $J$.

By definition of $\FF$:
 * $J \subseteq I$

By definition of set equality:
 * $I = J$

Hence:
 * $h'= h \in U$

Since $U$ was arbitrary, by definition of convergance:
 * the generalized sum $\ds \sum_{i \mathop \in I} g_i$

converges to:
 * the summation over finite index $\ds \sum_{i \mathop \in I} g_i$