Finite Generalized Sum Converges to Summation
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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 $\ds \sum_{i \mathop \in I} g_i$
$\blacksquare$