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I have been asked to prove the following and have been having some difficulty:

Let m,n and p be integers. If m<n then m+p<n+p

This is my proof thus far. It feels a bit sloppy and I'm also worried that my last part (indicated by ***) is wrong. I have not yet proved subtraction so don't think I can use it. Can anybody maybe check what I have so far and advise me on potential directions to explore? Thanks!

  1. By Prop 2.2 we know that all N must be greater than zero. We then can determine that n-m>0.
  2. By the additive inverse identity, we can obtain n-m+p-p>0
  3. By commutativity, we obtain n+p-m-p>0.
  4. ***Then, I desperately want to be able todo n+p>m+p, but I'm pretty sure this is incorrect since I haven't yet 'proved' subtraction. How do I get out of this? And also, am I on the right track?

Any insight would be greatly appreciated!

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    For $1$ you probably have to prove that $n-m\in\Bbb{N}$.Other then that you should probably mention what you've learned so far(what you can use). – 2017-02-05
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    Hmm. So far we have learned about addition and multiplication in the integers, and the definition of the natural numbers. The proof I am working on comes from a section on ordering the natural numbers. – 2017-02-05

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Hint (since it's not obvious what properties you have proved already, and are allowed to use): for a minimalistic proof, consider induction in $p$ to prove it for $p \ge 0$. Then use $m+p \lt n+p$ $\iff$ $(-n)+(-p) \lt (-m)+(-p)$ to prove it for negative $p$ using the previous step.

For $p=0$ the inequality reduces to the premise $m \lt n$, thus holds.

Assuming it holds for $p \ge 0$ that means $m+p \lt n+p$. But in integer numbers:

  • $a \lt b$ $\iff$ $a+1 \le b$ (since the next larger integer than $a$ is $a+1$), so $m+p+1 \le n+p\,$;

  • $a+1 \le b$ $\iff$ $a+1 \lt b+1$ (since the next larger integer than $b$ is $b+1$), so $m+p+1 \lt n+p+1\,$, which completes the induction step.

The above proves the statement for $p \ge 0$. If $p \lt 0$ then use the already proven statement for $-p \gt 0$ with integers $-n \lt -m$ to obtain $(-n)+(-p) \lt (-m)+(-p)$ $\iff$ $m+p \lt n+p\,$.

(The only properties of integer inequalities that were used are: $\;a \lt b \iff a+1 \le b\;$ and: $\;a \lt b \iff -b \lt -a\;$.)