description/proof of that Cauchy sequence on metric space with induced topology has at most \(1\) accumulation value
Topics
About: metric space
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of Cauchy sequence on metric space.
- The reader knows a definition of accumulation value of net with directed index set.
Target Context
- The reader will have a description and a proof of the proposition that any Cauchy sequence on any metric space with the induced topology has at most \(1\) accumulation value.
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:
\(M\): \(\in \{\text{ the metric spaces }\}\), with the induced topology
\(s\): \(: \mathbb{N} \to M\), \(\in \{\text{ the Cauchy sequences }\}\)
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Statements:
\(\vert \{\text{ the accumulation values of } s\} \vert \le 1\)
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2: Proof
Whole Strategy: Step 1: suppose that there were \(2\) accumulation values, \(m, m' \in M\), and find a contradiction.
Step 1:
Let us suppose that there were some \(2\) accumulation values, \(m, m' \in M\), such that \(m \neq m'\).
\(d := dist (m, m')\), where \(0 \lt d\).
There would be the open balls, \(B_{m, d / 3}\) and \(B_{m', d / 3}\), which would be some neighborhoods of \(m\) and \(m'\).
There would be an \(n \in \mathbb{N}\) such that for each \(j, l \in \mathbb{N}\) such that \(n \lt j, l\), \(dist (s (j), s (l)) \lt d / 3\), because \(s\) was Cauchy.
There would be an \(j \in \mathbb{N}\) such that \(n \lt j\) and \(s (j) \in B_{m, d / 3}\), because \(m\) was an accumulation value.
For each \(l \in \mathbb{N}\) such that \(n \lt l\), \(d = dist (m', m) \le dist (m', s (l)) + dist (s (l), m)\), so, \(d - dist (s (l), m) \le dist (m', s (l))\).
\(dist (s (l), m) \leq dist (s (l), s (j)) + dist (s (j), m) \lt d / 3 + d / 3 = (2 / 3) d\).
\(d - (2 / 3) d = 1 / 3 d \lt dist (m', s (l))\).
So, \(s (l) \notin B_{m', d / 3}\).
So, \(m'\) would not be any accumulation value, a contradiction.
