description/proof of that for divergent non-negative double series on \(1\)-dimensional Euclidean metric space, series with sums order changed diverges
Topics
About: metric space
The table of contents of this article
Starting Context
Target Context
- The reader will have a description and a proof of the proposition that for any divergent non-negative double series on the \(1\)-dimensional Euclidean metric space, the series with the sums orders changed diverges.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\(J_1\): \(\subseteq \mathbb{N}\), such that \(J_1 \neq \emptyset\)
\(J_2\): \(\subseteq \mathbb{N}\), such that \(J_2 \neq \emptyset\)
\(\mathbb{R}\): \(= \text{ the Euclidean metric space }\)
\(s\): \(: J_1 \times J_2 \to [0, \infty) \subseteq \mathbb{R}\), such that \(\sum_{j_1 \in J_1} \sum_{j_2 \in J_2} s (j_1, j_2) = \infty\)
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Statements:
\(\sum_{j_2 \in J_2} \sum_{j_1 \in J_1} s (j_1, j_2) = \infty\)
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2: Proof
Whole Strategy: Step 1: suppose that \(\sum_{j_2 \in J_2} s (j_1, j_2) = \infty\) for a \(j_1 \in J_1\); Step 2: see that for each \(r\), there is an \(N'\) such that for each \(N' \lt n'\), \(r \lt \sum_{j_2 \in \{{J_2}_1, ..., {J_2}_{n'}\}} \sum_{j_1 \in J_1} s (j_1, j_2)\); Step 3: suppose that \(\sum_{j_2 \in J_2} s (j_1, j_2) \in \mathbb{R}\) for each \(j_1 \in J_1\); Step 4: see that for each \(r\), \(r + \epsilon \lt \sum_{j_1 \in \{{J_1}_1, ..., {J_1}_N\}} \sum_{j_2 \in J_2} s (j_1, j_2)\), there is an \(N'\) such that for each \(N' \lt n'\), \(\sum_{j_2 \in J_2} s (j_1, j_2) - \sum_{j_2 \in \{{J_2}_1, ..., {J_2}_{n'}\}} s (j_1, j_2) \lt \epsilon / N\), and \(r \lt \sum_{j_1 \in \{{J_1}_1, ..., {J_1}_N\}} \sum_{j_2 \in \{{J_2}_1, ..., {J_2}_{n'}\}} s (j_1, j_2) = \sum_{j_2 \in \{{J_2}_1, ..., {J_2}_{n'}\}} \sum_{j_1 \in \{{J_1}_1, ..., {J_1}_N\}} s (j_1, j_2)\), and \(r \lt \sum_{j_2 \in \{{J_2}_1, ..., {J_2}_{n'}\}} \sum_{j_1 \in J_1} s (j_1, j_2)\).
Step 1:
Let us suppose that \(\sum_{j_2 \in J_2} s (j_1, j_2) = \infty\) for a \(j_1 \in J_1\).
Step 2:
Let \(r \in \mathbb{R}\) be any.
There is an \(N' \in \mathbb{N}\) such that for each \(n' \in \mathbb{N}\) such that \(N' \lt n'\), \(r \lt \sum_{j_2 \in \{{J_2}_1, ..., {J_2}_{n'}\}} s (j_1, j_2)\).
As \(0 \le s (j_1, j_2)\) for each \(j_1 \in J_1\) and \(j_2 \in J_2\), \(s (j_1, j_2) \le \sum_{j_1 \in J_1} s (j_1, j_2)\) for each \(j_2 \in J_2\).
So, \(r \lt \sum_{j_2 \in \{{J_2}_1, ..., {J_2}_{n'}\}} s (j_1, j_2) \le \sum_{j_2 \in \{{J_2}_1, ..., {J_2}_{n'}\}} \sum_{j_1 \in J_1} s (j_1, j_2)\).
That means that \(\sum_{j_2 \in J_2} \sum_{j_1 \in J_1} s (j_1, j_2) = \infty\).
Step 3:
Let us suppose that \(\sum_{j_2 \in J_2} s (j_1, j_2) \in \mathbb{R}\) for each \(j_1 \in J_1\).
Step 4:
Let \(r \in \mathbb{R}\) be any.
Let \(\epsilon \in \mathbb{R}\) be any such that \(0 \lt \epsilon\).
There is an \(N \in \mathbb{N}\) such that \(r + \epsilon \lt \sum_{j_1 \in \{{J_1}_1, ..., {J_1}_N\}} \sum_{j_2 \in J_2} s (j_1, j_2)\).
For each \(j_1 \in \{{J_1}_1, ..., {J_1}_N\}\), there is an \(N'_{j_1} \in \mathbb{N}\) such that for each \(n' \in \mathbb{N} \setminus \{0\}\) such that \(N'_{j_1} \lt n'\), \(\sum_{j_2 \in J_2} s (j_1, j_2) - \sum_{j_2 \in \{{J_2}_1, ..., {J_2}_{n'}\}} s (j_1, j_2) \lt \epsilon / N\).
Let \(N' := Max (\{N'_{{J_1}_1}, ..., N'_{{J_1}_N}\})\).
For each \(n' \in \mathbb{N} \setminus \{0\}\) such that \(N' \lt n'\), \(\sum_{j_2 \in J_2} s (j_1, j_2) - \sum_{j_2 \in \{{J_2}_1, ..., {J_2}_{n'}\}} s (j_1, j_2) \lt \epsilon / N\) for each \(j_1 \in \{{J_1}_1, ..., {J_1}_N\}\).
\(r + \epsilon \lt \sum_{j_1 \in \{{J_1}_1, ..., {J_1}_N\}} \sum_{j_2 \in J_2} s (j_1, j_2) = \sum_{j_1 \in \{{J_1}_1, ..., {J_1}_N\}} (\sum_{j_2 \in J_2} s (j_1, j_2) - \sum_{j_2 \in \{{J_2}_1, ..., {J_2}_{n'}\}} s (j_1, j_2) + \sum_{j_2 \in \{{J_2}_1, ..., {J_2}_{n'}\}} s (j_1, j_2)) \lt \sum_{j_1 \in \{{J_1}_1, ..., {J_1}_N\}} (\epsilon / N + \sum_{j_2 \in \{{J_2}_1, ..., {J_2}_{n'}\}} s (j_1, j_2)) = \sum_{j_1 \in \{{J_1}_1, ..., {J_1}_N\}} \epsilon / N + \sum_{j_1 \in \{{J_1}_1, ..., {J_1}_N\}} \sum_{j_2 \in \{{J_2}_1, ..., {J_2}_{n'}\}} s (j_1, j_2) = \epsilon + \sum_{j_1 \in \{{J_1}_1, ..., {J_1}_N\}} \sum_{j_2 \in \{{J_2}_1, ..., {J_2}_{n'}\}} s (j_1, j_2)\).
But \(\sum_{j_1 \in \{{J_1}_1, ..., {J_1}_N\}} \sum_{j_2 \in \{{J_2}_1, ..., {J_2}_{n'}\}} s (j_1, j_2) = \sum_{j_2 \in \{{J_2}_1, ..., {J_2}_{n'}\}} \sum_{j_1 \in \{{J_1}_1, ..., {J_1}_N\}} s (j_1, j_2)\), by the proposition that for any absolutely convergent double series on the \(1\)-dimensional Euclidean metric space, the series with the sums orders changed converge to the same convergence.
So, \(r \lt \sum_{j_1 \in \{{J_1}_1, ..., {J_1}_N\}} \sum_{j_2 \in \{{J_2}_1, ..., {J_2}_{n'}\}} s (j_1, j_2) = \sum_{j_2 \in \{{J_2}_1, ..., {J_2}_{n'}\}} \sum_{j_1 \in \{{J_1}_1, ..., {J_1}_N\}} s (j_1, j_2)\).
As \(0 \le s (j_1, j_2)\) for each \(j_1 \in J_1\) and \(j_2 \in J_2\), \(\le \sum_{j_2 \in \{{J_2}_1, ..., {J_2}_{n'}\}} \sum_{j_1 \in J_1} s (j_1, j_2)\).
So, \(r \lt \sum_{j_2 \in \{{J_2}_1, ..., {J_2}_{n'}\}} \sum_{j_1 \in J_1} s (j_1, j_2)\).
That means that \(\sum_{j_2 \in J_2} \sum_{j_1 \in J_1} s (j_1, j_2) = \infty\).
