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I am reading Ethan D. Bloch's book: The Real Numbers and Real Analysis ...
I am currently focused on Chapter 1: Construction of the Real Numbers ...
I need help/clarification with an aspect of Theorem 1.2.7 (1) ...
Theorem 1.2.7 reads as follows:
https://www.physicsforums.com/attachments/6976
https://www.physicsforums.com/attachments/6977
In the above proof of (1) we read the following:" We will show that $$G = \mathbb{N}$$, which will imply the desired result. Clearly $$G \subseteq \mathbb{N}$$. ... ... ... "Before he proves that $$1 \in G$$, Bloch asserts that $$G \subseteq \mathbb{N}$$ ... what is his reasoning ...?
It does not appear to me ... from the order in which he says things that he is saying
$$1 \in G$$ ... therefore $$G \subseteq \mathbb{N}$$ ...
Can we immediately conclude that $$G \subseteq \mathbb{N}$$ without relying on $$1 \in G $$... ... ?Hope someone can help ... ...
Peter
I am currently focused on Chapter 1: Construction of the Real Numbers ...
I need help/clarification with an aspect of Theorem 1.2.7 (1) ...
Theorem 1.2.7 reads as follows:
https://www.physicsforums.com/attachments/6976
https://www.physicsforums.com/attachments/6977
In the above proof of (1) we read the following:" We will show that $$G = \mathbb{N}$$, which will imply the desired result. Clearly $$G \subseteq \mathbb{N}$$. ... ... ... "Before he proves that $$1 \in G$$, Bloch asserts that $$G \subseteq \mathbb{N}$$ ... what is his reasoning ...?
It does not appear to me ... from the order in which he says things that he is saying
$$1 \in G$$ ... therefore $$G \subseteq \mathbb{N}$$ ...
Can we immediately conclude that $$G \subseteq \mathbb{N}$$ without relying on $$1 \in G $$... ... ?Hope someone can help ... ...
Peter