Generalized Sum with Countable Non-zero Summands/Corollary

Theorem
Let $V$ be a Banach space.

Let $\norm {\, \cdot \,}$ denote the norm on $V$.

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

Let $J, K$ be countably infinite subsets of $I$ such that $\set{i \in I : v_i \ne 0} \subseteq J \cap K$.

Let $\set{j_0, j_1, j_2, \ldots}$ and $\set{k_0, k_1, k_2, \ldots}$ be countably infinite enumerations of $J$ and $K$ respectively.

Let $r \in \R_{\mathop > 0}$.

Then:
 * the series $\ds \sum_{n \mathop = 1}^\infty v_{j_n}$ converges absolutely to $r$


 * the series $\ds \sum_{n \mathop = 1}^\infty v_{k_n}$ converges absolutely to $r$
 * the series $\ds \sum_{n \mathop = 1}^\infty v_{k_n}$ converges absolutely to $r$

Proof
From Generalized Sum with Countable Non-zero Summands:
 * the series $\ds \sum_{n \mathop = 1}^\infty v_{j_n}$ converges absolutely to $r$


 * the generalized sum $\ds \sum_{i \mathop \in I} v_i$ converges absolutely to $r$


 * the series $\ds \sum_{n \mathop = 1}^\infty v_{k_n}$ converges absolutely to $r$
 * the series $\ds \sum_{n \mathop = 1}^\infty v_{k_n}$ converges absolutely to $r$