Generalized Sum Restricted to Non-zero Summands

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

Let $g'= g \restriction_J$ be the restriction of $g$ to $J$.

Let $h \in G$.

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


 * the generalized sum $\ds \sum_{j \mathop \in J} g'_j$ converges to $h$
 * the generalized sum $\ds \sum_{j \mathop \in J} g'_j$ converges to $h$

Necessary Condition
Let the generalized sum $\ds \sum_{i \mathop \in I} g_i$ converge to $h$.

It will be shown that $\ds \sum_{j \mathop \in J} g'_j$ converges to $h$.

Sufficient Condition
Let the generalized sum $\ds \sum_{j \mathop \in J} g'_j$ converge to $h \in G$.

It will be shown that $\ds \sum_{i \mathop \in I} g_j$ converges to $h$.