1
$\begingroup$

I have been trying to prove Archimedean property ($n\cdot x>y$) on my own and came up with this.

No matter how big $y$ is given, I can get next integer of that value of $y$. And no matter how small $x$ is, I can get a terminating decimal in base 10, which is rational and then I will make my $n$ as denominator of that rational times the next integer of $y$, which will give $n\cdot x>y$. Is this valid or have I missed something??

  • 0
    The Archimedean property is one of the defining properties of $\Bbb R$ (at least with the definition I use). Are you trying to prove that *a particular model* of $\Bbb R$ has that property? – 2017-02-21
  • 0
    The one proven in rudin's book on real analysis. – 2017-02-21
  • 2
    What's your definition of $\Bbb{R}$? – 2017-02-21
  • 1
    @jnyan For those of us who don't have a copy of PMA sitting within reach, could you edit into the question your definition of $\Bbb R$? – 2017-02-21
  • 0
    The Archimedean property is already present in ${\mathbb N}$. A proof would have to start from the Peano axioms. – 2017-02-21

1 Answers 1

1

I think you are assuming the theorem in your proof, more or less. Regardless you are not showing how you get the rational number you use in your proof.

Assuming you are working with $\mathbb{R}$ as $\mathbb{Q}$ with supremum for nonempty sets bounded above, suppose no such integer exists. Thus $\Bbb Z$ is bounded above by $\frac{y}{x}$, and then has a supremum $s$. It follows that since $s-1$ is less than the supremum of $\Bbb Z$, there exists some $n\in \Bbb Z$ such that $s-1< n$, and therefore $s< n+1$, a contradiction.

  • 0
    I never used the fact that $\Bbb N$ has the A.P., only that the adding one does not change the inequality, something which is true once we order the reals. – 2017-02-21
  • 0
    Sorry. I had misplaced my comment. – 2017-02-21
  • 0
    any real number will have a decimal expansion. say like 0.1234....i will use a smaller decimal number, like 0.12 which is rational. then i can take n as product of denominator of rational form of 0.12 and next integer of y. I dont see anything wrong, or using the proof to prove my proof. – 2017-02-22
  • 0
    How do you know every real number has a decimal expansion? How do you know there is a smaller one? You are glossing over these points. They might seem obvious, but to be fair so does the arch. Prop. – 2017-02-22