Generalized Sum Restricted to Non-zero Summands/Necessary Condition
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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 $J = \set{i \in I : g_i \ne 0_G}$
Let $h \in G$.
Let the generalized sum $\ds \sum_{i \mathop \in I} g_i$ converge to $h$.
Then:
- the generalized sum $\ds \sum_{j \mathop \in J} g_j$ converges to $h$
Proof
Let $U \subseteq G$ be an open subset of $G$ such that $h \in U$.
By definition of convergent net:
- $(1) \quad \exists F \subseteq I : F \ne \O : \forall E \subseteq I : E \supseteq F \implies \ds \sum_{i \mathop \in E} g_i \in U$
where $\ds \sum_{i \mathop \in E} g_i$ is the summation over $E$.
Let:
- $F'= F \cap J$
From Set Difference and Intersection form Partition:
- $F = F' \cup F \setminus J$
Let $E' \subseteq J$:
- $E' \supseteq F'$
We have:
- $E' \cap F \setminus J = \O$
Let:
- $E = E' \cup F \setminus J$
From Set Union Preserves Subsets:
- $E \supseteq F$
From $(1)$:
- $\ds \sum_{i \mathop \in E} g_i \in U$
Case : $F \setminus J = \O$
Let:
- $F \setminus J = \O$
From Union with Empty Set:
- $E = E'$
Hence:
- $\ds \sum_{i \mathop \in E'} g_i \in U$
$\Box$
Case : $F \setminus J \ne \O$
Let:
- $F \setminus J \ne \O$
We have:
| \(\ds \sum_{i \mathop \in E} g_i\) | \(=\) | \(\ds \sum_{i \mathop \in E'} g_i + \sum_{i \mathop \in F \setminus J} g_i\) | Summation over Union of Disjoint Finite Index Sets | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sum_{i \mathop \in E'} g_i + \sum_{i \mathop \in F \setminus J} 0_G\) | definitions of $F$ and $J$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sum_{i \mathop \in E'} g_i\) | Definition of identity |
Hence:
- $\ds \sum_{i \mathop \in E'} g_i \in U$
$\Box$
In either case:
- $\ds \sum_{i \mathop \in E'} g_i \in U$
Since $U$ was arbitrary, it follows that $\ds \sum_{j \mathop \in J} g_j$ converges to $h$ by definition.
$\blacksquare$