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I have the following proposition: If $p > 0$ and $mp < np$, then $m < n$. Proof: $p\in\mathbb N$. \begin{align*} np - mp \in\mathbb N\\ p\cdot(n-m) \in\mathbb N \end{align*} If $p$ and $p \cdot(n-m)$ both $\in\mathbb N$, then $(n - m) \in\mathbb N$ because if $n - m \notin\mathbb N$, then the product wouldn't be a natural number. Hence, $m < n$.

What do you think? Should I try to get rid of the p instead? Thank you!

Johnathan
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  • This problem depends greatly on what you were given as tools to use. In general, it is not correct to conclude that if $ab\in\mathbb{N}$, then $a,b\in\mathbb{N}$; e.g. $a=b=-1$. Or take $a=8, b=\frac{1}{2}$. – vadim123 Mar 04 '15 at 02:54
  • @vadim123 The OP was not only given $ab \in \mathbb{N}$, but also that one of the factors was also $a \in \mathbb{N}$, given we are only dealing with integers. So in that case it is perfectly correct. – Machine Caliber Mar 04 '15 at 02:56
  • @OrangeSleipnir, that is incorrect, without additional assumptions. – vadim123 Mar 04 '15 at 02:56
  • @vadim123 Would you like to say what part of that is incorrect? – Machine Caliber Mar 04 '15 at 02:56
  • You're right, which is why I was going to post an answer to this. In my comment I was only arguing for the integer case, I'll make that clear. – Machine Caliber Mar 04 '15 at 02:58
  • Technically there's nothing wrong, but it somewhat worries me that you're using subtraction and the distributive property without having proved this basic cancellation fact; you also seem to be using properties about integers that might not have been proven before. For a most basic proof (in the spirit of Peano arithmetic), I would suggest induction on $p$. – Pedro M. Mar 04 '15 at 02:58

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First, $mp < np \Leftrightarrow np - mp > 0 \Leftrightarrow (n-m)p > 0$. It seems unclear to me why you are using $np-mp \in \mathbb{N}$ rather than the basic definition of the inequality. After the last step above, you can simply use the fact that $ab > 0, a > 0 \Rightarrow b > 0$. (Or if you can't use that, you can simply prove it.)

If you are asking for proof style, I would personally think that if you will be resorting to a proof by contradiction, then you should just start with that tactic from the beginning. For example, you could start by supposing that $m \geq n$, and show that $mp \geq np$ in this case.

Machine Caliber
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