description/proof of that for \(2\) convergent sequences with same domain on \(1\)-dimensional Euclidean metric space, sequence with elements as sums of corresponding elements converges with sum of convergences
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 \(2\) convergent sequences with any same domain on the \(1\)-dimensional Euclidean metric space, the sequence with the elements as the sums of the corresponding elements converges with the sum of the convergences.
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 \(J \neq \emptyset\)
\(\mathbb{R}\): \(= \text{ the Euclidean metric space }\)
\(s_1\): \(: J \to \mathbb{R}\), such that \(lim s_1 = r_1 \in \mathbb{R}\)
\(s_2\): \(: J \to \mathbb{R}\), such that \(lim s_2 = r_2 \in \mathbb{R}\)
\(s_1 + s_2\): \(: J \to \mathbb{R}, j \mapsto s_1 (j) + s_2 (j)\)
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Statements:
\(lim (s_1 + s_2) = r_1 + r_2 \in \mathbb{R}\)
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2: Proof
Whole Strategy: Step 1: deal with the case that \(J\) is finite, and suppose otherwise, thereafter; Step 2: for each \(\epsilon\), take an \(N_1\) such that for each \(N_1 \lt n\), \(\vert r_1 - s_1 (J_n) \vert \lt \epsilon / 2\) and an \(N_2\) such that for each \(N_2 \lt n\), \(\vert r_2 - s_2 (J_n) \vert \lt \epsilon / 2\); Step 3: see that for each \(N_1, N_2 \lt n\), \(\vert r_1 + r_2 - (s_1 + s_2) (J_n) \vert \lt \epsilon\).
Step 1:
When \(\vert J \vert = n \in \mathbb{N}\), it holds, because \(lim (s_1 + s_2) = (s_1 + s_2) (J_n) = s_1 (J_n) + s_2 (J_n) = r_1 + r_2\).
Let us suppose otherwise, hereafter.
Step 2:
Let \(\epsilon \in \mathbb{R}\) be any such that \(0 \lt \epsilon\).
There is an \(N_1 \in \mathbb{N}\) such that for each \(n \in \mathbb{N}\) such that \(N_1 \lt n\), \(\vert r_1 - s_1 (J_n) \vert \lt \epsilon / 2\), by the definition of convergence.
There is an \(N_2 \in \mathbb{N}\) such that for each \(n \in \mathbb{N}\) such that \(N_2 \lt n\), \(\vert r_2 - s_2 (J_n) \vert \lt \epsilon / 2\), by the definition of convergence.
Step 3:
For each \(n \in \mathbb{N}\) such that \(N_1, N_2 \lt n\), \(\vert r_1 + r_2 - (s_1 + s_2) (J_n) \vert = \vert r_1 - s_1 (J_n) + r_2 - s_2 (J_n) \vert \le \vert r_1 - s_1 (J_n) \vert + \vert r_2 - s_2 (J_n) \vert \lt \epsilon / 2 + \epsilon / 2 = \epsilon\).
That means that \(lim (s_1 + s_2) = r_1 + r_2 \in \mathbb{R}\).
