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 subset of $V$.

Let $\set{i \in I : v_i \ne 0}$ be countably infinite. Let $\set{i_0, i_1, i_2, \ldots}$ and $\set{j_0, j_1, j_2, \ldots}$ be countable infinite enumerations of $\set{i \in I : v_i \ne 0}$.

Then:
 * the series $\ds \sum_{n = 1}^\infty v_{i_n}$ converges absolutely


 * the series $\ds \sum_{n = 1}^\infty v_{j_n}$ converges absolutely
 * the series $\ds \sum_{n = 1}^\infty v_{j_n}$ converges absolutely

In which case:
 * $\ds \sum_{n = 1}^\infty \norm{v_{i_n}} = \sum_{n = 1}^\infty \norm{v_{j_n}}$

Proof
From User:Leigh.Samphier/Topology/Generalized Sum with Countable Non-zero Summands:
 * the series $\ds \sum_{n = 1}^\infty v_{i_n}$ converges absolutely


 * the generalized sum $\ds \sum_{i \in I} g_i$ converges absolutely


 * the series $\ds \sum_{n = 1}^\infty v_{j_n}$ converges absolutely
 * the series $\ds \sum_{n = 1}^\infty v_{j_n}$ converges absolutely

In which case:
 * $\ds \sum_{n = 1}^\infty \norm{v_{i_n}} = \sum_{i \in I} \norm{g_i} = \sum_{n = 1}^\infty \norm{v_{j_n}}$