Prove that addition preserves order. (for natural numbers)

Question

Prove that addition preserves order. $a ≥ b$ if and only if $a+c ≥ b+c$. (using peano axioms)
I try to do it by induction on $c$. Can I use $(a+c)++ ≥ (b+c)++$. I am not sure because first we will have to prove that if $a\geq b$ then $a++\geq b++$ how do I do that ?

Answer 1

The step you have trouble with (namely, the case $c=1$) amounts to the fact that the sum is commutative.

If $b\le a$, then there exists $d$ such that $a=b+d$.

Hence $a+1=(b+d)+1=(b+1)+d$.

Once you have case $c=1$, you can indeed use it for the other inductive steps.

Observation: You can do the proof above for a general $c$, actually, and spare yourself the induction.