Theorem: Absolute Net Convergence Equivalent to Absolute Convergence/Absolute Convergence implies Absolute Net Convergence

Let $V$ be a Banach space.
Let $\sequence {v_n}_{n \mathop \in \N}$ be a sequence of elements in $V$.
Let $r \in \R_{\mathop \ge 0}$
Let the series $\ds \sum_{n \mathop = 1}^\infty v_n$ be absolutely convergent to $r$.
Then:
:the generalized sum $\ds \sum \set {v_n: n \in \N}$ is absolutely net convergent to $r$.

Proof:
Let $\epsilon \in \R_{\mathop \ge 0}$.
By definition of absolutely convergent:
:$(2) \quad \exists N \in \N : \forall m \ge N : \size{\ds \sum_{n \mathop = 0}^m \norm{v_m} - r} < \dfrac \epsilon 3$
Let:
:$F = \closedint 0 N$
Let:
:$E \subseteq \N : E \supseteq F : E$ is finite.
Let:
:$m = \max \set{n : n \in E}$
Let:
:$G = \closedint 0 m$
We have:
:$F = \closedint 0 N \subseteq E \subseteq \closedint 0 m = G$
From Set Difference and Intersection form Partition:
:$E = F \cup E \setminus F$
and
:$G = F \cup G \setminus F$
From Set Difference Intersection with Second Set is Empty Set:
:$F \cap E \setminus F = \O$
and
:$F \cap G \setminus F = \O$
From Set Difference over Subset:
:$E \setminus F \subseteq G \setminus F$
We have:
{{begin-eqn}}
{{eqn | l = \size {\sum_{n \mathop \in E} \norm {v_n} - r}
      | r = \size{\sum_{n \mathop \in F} \norm {v_n} + \sum_{n \mathop \in E \setminus F} \norm {v_n} - r }
      | c = Summation over Union of Disjoint Finite Index Sets
}}
{{eqn | o = \le
      | r = \size{\sum_{n \mathop \in F} \norm {v_n} - r } + \size{ \sum_{n \mathop \in E \setminus F} \norm {v_n} }
      | c = Triangle Inequality for Real Numbers
}}
{{eqn | o = \le
      | r = \size{\sum_{n \mathop \in F} \norm {v_n} - r } + \size{ \sum_{n \mathop \in G \setminus F} \norm {v_n} }
}}
{{eqn | r = \size{\sum_{n \mathop \in F} \norm {v_n} - r } + \size{ \sum_{n \mathop \in G} \norm {v_n} - \sum_{n \mathop \in F} \norm {v_n} }
      | c = Summation over Union of Disjoint Finite Index Sets
}}
{{eqn | r = \size{\sum_{n \mathop \in F} \norm {v_n} - r } + \size{ \sum_{n \mathop \in G} \norm {v_n} -r + r - \sum_{n \mathop \in F} \norm {v_n} }
}}
{{eqn | o = \le
      | r = \size{\sum_{n \mathop \in F} \norm {v_n} - r } + \size{ \sum_{n \mathop \in G} \norm {v_n} -r } + \size{ r - \sum_{n \mathop \in F} \norm {v_n} }
      | c = Triangle Inequality for Real Numbers
}}
{{eqn | r = \size{\sum_{n \mathop = 0}^N \norm {v_n} - r } + \size{ \sum_{n \mathop = 0}^m \norm {v_n} -r } + \size{ r - \sum_{n \mathop = 0}^N \norm {v_n} }
      | c = {{Defof|Summation over Finite Index}}
}}
{{eqn | o = <
      | r = \dfrac \epsilon 3 + \dfrac \epsilon 3 + \dfrac \epsilon 3
      | c = from $(2)$
}}
{{eqn | r = \epsilon
}}
{{end-eqn}}
Since $E$ was arbitrary, it follows:
:$\exists F \subset \N: F $ is finite $: \forall E \subseteq \N : E \supseteq F: E$ is finite $\leadsto \size{\ds \sum_{n \mathop \in E} \norm{v_n} - r} < \epsilon$
Sine $\epsilon$ was arbitrary, it follows:
:$\forall \epsilon \in \R_{\mathop > 0} : \exists F \subset \N: F $ is finite $: \forall E \subseteq \N : E \supseteq F: E$ is finite $\leadsto \size{\ds \sum_{n \mathop \in E} \norm{v_n} - r} < \epsilon$
From Characterization of Convergent Net in Metric Space:
:the generalized sum $\ds \sum \set {v_n: n \in \N}$ is absolutely net convergent to $r$.