description/proof of that sum of uncountable number of positive real numbers is infinity
Topics
About: metric space
The table of contents of this article
Starting Context
- The reader knows a definition of sum of uncountable number of non-negative real numbers.
Target Context
- The reader will have a description and a proof of the proposition that the sum of any uncountable number of positive real numbers is infinity.
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:
\(J\): \(\in \{\text{ the uncountable index sets }\}\)
\( \mathbb{R}\): \(= \text{ the Euclidean metric space }\) with the canonical linear ordering
\(\{r_j \in (0, \infty) \subseteq \mathbb{R} \vert j \in J\}\):
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Statements:
\(\sum_{j \in J} r_j = \infty\)
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2: Proof
Whole Strategy: Step 1: see that for each \(M \in \mathbb{R}\), there is a countable subset, \(J^` \subseteq J\), such that \(M \lt \sum_{j \in J^`} r_j\).
Step 1:
\(\sum_{j \in J} r_j = \infty\) means that there is no supremum of \(\{\sum_{j \in J^`} r_j \vert J^` \in \{\text{ the countable subsets of } J\}\}\) in \(\mathbb{R}\), which is colloquially expressed as "the supremum is \(\infty\)".
That exactly means that for each \(M \in \mathbb{R}\), there is a countable subset, \(J^` \subseteq J\), such that \(M \lt \sum_{j \in J^`} r_j\).
Let \(M \in \mathbb{R}\) be any.
For each \(n \in \mathbb{N} \setminus \{0\}\), let us take \(J_n := \{j \in J \vert 1 / n \lt r_j\}\).
\(J_n\) may be finite or infinite.
Let us see that \(J_n\) cannot be finite for each \(n\).
\(J = \cup_{n \in \mathbb{N} \setminus \{0\}} J_n\), because for each \(j \in J\), \(1 / n \lt r_j\) for an \(n\), so, \(j \in J_n\), so, \(J \subseteq \cup_{n \in \mathbb{N} \setminus \{0\}} J_n\), while \(\cup_{n \in \mathbb{N} \setminus \{0\}} J_n \subseteq J\) is obvious.
If \(J_n\) was finite for each \(n\), \(J\) would be countable as a countable union of finite sets, a contradiction, so, there is at least \(1\) infinite \(J_n\).
Let us suppose that \(J_n\) is infinite.
When \(J_n\) is uncountable, there is an infinite countable subset of it, denoted as \(J_n\) again.
\(\vert J_n \vert / n = \sum_{j \in J_n} 1 / n \lt \sum_{j \in J_n} r_j\).
But the left hand side diverges, so, the right hand side diverges, so, \(M \lt \sum_{j \in J_n} r_j\).
So, there is a \(J^` := J_n\) such that \(M \lt \sum_{j \in J^`} r_j\).
So, \(\sum_{j \in J} r_j = \infty\).
