0

I'm trying to understand the solution to the following problem I took from Hijab's "Introduction to Calculus and Analysis":

enter image description here

The solution is:

enter image description here enter image description here

We are using the following definition and theorem:

enter image description here

enter image description here

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 $n\in S$?

Marius S.L.
  • 2,245
Red Banana
  • 23,956
  • 20
  • 91
  • 192
  • 3
    $S$ is literally defined as the set of naturals $n$ such that $n < r < n+1$ is not possible for any natural $r$. So $n < m - 1 < n+1$ contradicts that with $r = m - 1$ – balddraz Jun 03 '23 at 00:30
  • 2
    It's recommended to type in the mathematics in latex format rather than use images on this site. I'm surprised such a high reputation user would enter text as images. – Suzu Hirose Jun 03 '23 at 00:35
  • @OXLR - Do you have a reference for the definition that you stated? – uniquesolution Jun 03 '23 at 00:43

1 Answers1

1

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 $n=2,3,4,...$.

Suzu Hirose
  • 11,660