Generalized Sum Restricted to Non-zero Summands/Sufficient 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 $\ds \sum_{j \mathop \in J} g_j$ converge to $h$.


Then:

the generalized sum $\ds \sum_{i \mathop \in I} g_i$ converges to $h$.

Proof

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

By definition of convergent net:

$(2) \quad \exists F' \subseteq J : F' \ne \O : \forall E' \subseteq J : E' \supseteq F' \implies \ds \sum_{j \mathop \in E'} g_j \in U$

where $\ds \sum_{j \mathop \in E'} g_j$ is the summation over $E$.

We have:

$F' \subseteq J \subseteq I$.


Let $E \subseteq I$:

$E \supseteq F'$


Let:

$E' = E \cap J$

From Set Intersection Preserves Subsets and Intersection with Subset is Subset:

$E' \supseteq F'$

From $(2)$:

$\ds \sum_{j \mathop \in E'} g_j \in U$


From Set Difference Union Intersection:

$E = E' \cup E \setminus J$

From Set Difference and Intersection are Disjoint:

$E' \cap E \setminus J = \O$

Case : $E \setminus J = \O$

Let:

$E \setminus J = \O$

From Union with Empty Set:

$E = E'$

Hence:

$\ds \sum_{i \mathop \in E} g_i \in U$

$\Box$

Case : $E \setminus J \ne \O$

Let:

$E \setminus J \ne \O$

We have:

\(\ds \sum_{i \mathop \in E} g_i\) \(=\) \(\ds \sum_{i \mathop \in E'} g_i + \sum_{i \mathop \in E \setminus J} g_i\) Summation over Union of Disjoint Finite Index Sets
\(\ds \) \(=\) \(\ds \sum_{i \mathop \in E'} g_i + \sum_{i \mathop \in E \setminus J} 0_G\) definitions of $E$ 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 $\ds \sum_{i \mathop \in I} g_i$ converges to $h$ by definition.

$\blacksquare$

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