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

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SUMMARY

The discussion focuses on Lemma 1.2.3 from Ethan D. Bloch's "The Real Numbers and Real Analysis." It establishes that for any element ##n## in a set ##G##, if the successor function ##s(n)## yields an element ##p##, then ##p## is also in ##G##. This leads to the conclusion that the natural numbers ##\mathbb{N}## are a subset of ##G##, confirming that both 1 and all successors of elements in ##G## belong to ##G##.

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anhtudo
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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.
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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##.
 
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Thank you.
 

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