anhtudo
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- TL;DR
- What I don't understand is how he proves that G = N.
I don't think it is logical to let b = n as it can not be derived from the definition of G that b is in G.
Thanks.
Lemma 1.2.3 - Ethan.D.Bloch - The Real Numbers and Real Analysis
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It is written a bit confusing, but correct. Forget ##b##.
We have ##n\in G## and ##p=s(n)##. This implies ##p \in G## because there is some ##n \in \mathbb{N}## such that ##s(n)=p##. Hence ##s(n) \in G##. Therefore we have ##1\in G## and all successors of elements of ##G## are in ##G##, too, i.e. ##\mathbb{N} \subseteq G##.
We have ##n\in G## and ##p=s(n)##. This implies ##p \in G## because there is some ##n \in \mathbb{N}## such that ##s(n)=p##. Hence ##s(n) \in G##. Therefore we have ##1\in G## and all successors of elements of ##G## are in ##G##, too, i.e. ##\mathbb{N} \subseteq G##.
anhtudo
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Thank you.