description/proof of that for convergent sequence on \(1\)-dimensional Euclidean metric space and real number, sequence with elements as corresponding elements multiplied by number converges with convergence multiplied by number
Topics
About: metric space
The table of contents of this article
Starting Context
- The reader knows a definition of convergence of sequence on metric space.
Target Context
- The reader will have a description and a proof of the proposition that for any convergent sequence on the \(1\)-dimensional Euclidean metric space and any real number, the sequence with the elements as the corresponding elements multiplied by the number converges with the convergence multiplied by the number.
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:
\(J\): \(\subseteq \mathbb{N}\), such that \(J \neq \emptyset\)
\(\mathbb{R}\): \(= \text{ the Euclidean metric space }\)
\(s\): \(: J \to \mathbb{R}\), such that \(lim s = r \in \mathbb{R}\)
\(r'\): \(\in \mathbb{R}\)
\(r' s\): \(: J \to \mathbb{R}, j \mapsto r' s (j)\)
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Statements:
\(lim r' s = r' r \in \mathbb{R}\)
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2: Proof
Whole Strategy: Step 1: deal with the case that \(J\) is finite and with the case that \(r' = 0\), and suppose otherwise, thereafter; Step 2: for each \(\epsilon\), take an \(N\) such that for each \(N \lt n\), \(\vert r - s (J_n) \vert \lt \epsilon / \vert r' \vert\); Step 3: see that for each \(N \lt n\), \(\vert r' r - r' s (J_n) \vert \lt \epsilon\).
Step 1:
When \(\vert J \vert = n \in \mathbb{N}\), it holds, because \(lim r' s = r' s (J_n) = r' r\).
Let us suppose otherwise, hereafter.
Let us suppose that \(r' = 0\).
For each \(j \in J\), \(r' s (j) = 0 s (j) = 0\), so, \(lim r' s = 0\).
\(r' r = 0 r = 0\).
So, \(lim r' s = r' r \in \mathbb{R}\).
Let us suppose that \(r' \neq 0\) hereafter.
Step 2:
Let \(\epsilon \in \mathbb{R}\) be any such that \(0 \lt \epsilon\).
There is an \(N \in \mathbb{N}\) such that for each \(n \in \mathbb{N}\) such that \(N \lt n\), \(\vert r - s (J_n) \vert \lt \epsilon / \vert r' \vert\), by the definition of convergence.
Step 3:
For each \(n \in \mathbb{N}\) such that \(N \lt n\), \(\vert r' r - r' s (J_n) \vert = \vert r' (r - s (J_n)) \vert = \vert r' \vert \vert r - s (J_n) \vert \lt \vert r' \vert \epsilon / \vert r' \vert = \epsilon\).
That means that \(lim r' s = r' r \in \mathbb{R}\).
