Generalized Sum over Subset of Absolutely Convergent Generalized Sum is Absolutely Convergent

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Theorem

Let $V$ be a Banach space.

Let $\family {v_i}_{i \mathop \in I}$ be an indexed family of elements of $V$.

Let the generalized sum $\ds \sum_{i \mathop \in I} v_i$ be absolutely net convergent.

Let $J \subseteq I$.

Then:

the generalized sum $\ds \sum_{j \mathop \in J} v_j$ is absolutely net convergent.

Proof

By definition of absolute net convergence, let:

$\ds \sum_{i \mathop \in I} \norm{v_i} = M$

Let $F \subseteq J$ be finite.

From Subset Relation is Transitive:

$F \subseteq I$

From Absolutely Convergent Generalized Sum Converges to Supremum:

$\ds \sum_{j \mathop \in F} \norm{v_j} \le M$

Since $F \subseteq J$ was arbitrary, it follows that:

$\forall F \subseteq J : F$ is finite $: \ds \sum_{j \mathop \in F} \norm{v_j} \le M$

From Bounded Generalized Sum is Absolutely Convergent:

$\ds \sum_{j \mathop \in J} \norm{v_j}$ is absolutely net convergent

$\blacksquare$