I'm trying to understand the solution to the following problem I took from Hijab's "Introduction to Calculus and Analysis":
The solution is:
We are using the following definition and theorem:
And when he says "from the text" is that he proved earlier in the text that there are no naturals between $1$ and $2$.
Question: Why does $n< m-1 < n+1$ contradicts $n\in S$?
If $n<m-1<n+1$, and $m-1\in\mathbf{N}$, then $n$ cannot be a member of the set $S$ for which there are no naturals between $n$ and $n+1$.
The point is to show that the property holds for $n=1$ and if it holds for $n$ then it holds for $n+1$, thus after inserting $n=1$ we get $n=2,3,4,...$.