I Lemma 1.2.3 - Ethan.D.Bloch - The Real Numbers and Real Analysis

[anhtudo]

TL;DR Summary

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.

 

[fresh_42]

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. $\mathbb{N} \subseteq G$.

 

[anhtudo]

Thank you.