Generalized Sum Restricted to Non-zero Summands/Corollary

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Theorem

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

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

Let $K \subseteq I : \set{i \in I : g_i \ne 0_G} \subseteq K$

Let $h \in G$.

Then:

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

if and only if:

the generalized sum $\ds \sum_{k \mathop \in K} g_k$ converges to $h$

Proof

Let $J = \set{i \in I : g_i \ne 0_G}$.

We have:

\(\ds \sum_{i \mathop \in I} g_i\)

\(\to\)

\(\ds h\)

\(\ds \leadstoandfrom \ \ \)

\(\ds \sum_{j \mathop \in J} g_j\)

\(\to\)

\(\ds h\)

Generalized Sum Restricted to Non-zero Summands

\(\ds \leadstoandfrom \ \ \)

\(\ds \sum_{k \mathop \in K} g_k\)

\(\to\)

\(\ds h\)

Generalized Sum Restricted to Non-zero Summands

$\blacksquare$