Absolute Net Convergence Equivalent to Absolute Convergence/Absolute Convergence implies Absolute Net Convergence
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Theorem
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:
\(\ds \size {\sum_{n \mathop \in E} \norm {v_n} - r}\) | \(=\) | \(\ds \size{\sum_{n \mathop \in F} \norm {v_n} + \sum_{n \mathop \in E \setminus F} \norm {v_n} - r }\) | Summation over Union of Disjoint Finite Index Sets | |||||||||||
\(\ds \) | \(\le\) | \(\ds \size{\sum_{n \mathop \in F} \norm {v_n} - r } + \size{ \sum_{n \mathop \in E \setminus F} \norm {v_n} }\) | Triangle Inequality for Real Numbers | |||||||||||
\(\ds \) | \(\le\) | \(\ds \size{\sum_{n \mathop \in F} \norm {v_n} - r } + \size{ \sum_{n \mathop \in G \setminus F} \norm {v_n} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \size{\sum_{n \mathop \in F} \norm {v_n} - r } + \size{ \sum_{n \mathop \in G} \norm {v_n} - \sum_{n \mathop \in F} \norm {v_n} }\) | Summation over Union of Disjoint Finite Index Sets | |||||||||||
\(\ds \) | \(=\) | \(\ds \size{\sum_{n \mathop \in F} \norm {v_n} - r } + \size{ \sum_{n \mathop \in G} \norm {v_n} -r + r - \sum_{n \mathop \in F} \norm {v_n} }\) | ||||||||||||
\(\ds \) | \(\le\) | \(\ds \size{\sum_{n \mathop \in F} \norm {v_n} - r } + \size{ \sum_{n \mathop \in G} \norm {v_n} -r } + \size{ r - \sum_{n \mathop \in F} \norm {v_n} }\) | Triangle Inequality for Real Numbers | |||||||||||
\(\ds \) | \(=\) | \(\ds \size{\sum_{n \mathop = 0}^N \norm {v_n} - r } + \size{ \sum_{n \mathop = 0}^m \norm {v_n} -r } + \size{ r - \sum_{n \mathop = 0}^N \norm {v_n} }\) | Definition of Summation over Finite Index | |||||||||||
\(\ds \) | \(<\) | \(\ds \dfrac \epsilon 3 + \dfrac \epsilon 3 + \dfrac \epsilon 3\) | from $(2)$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \epsilon\) |
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 $r$.
$\blacksquare$