Note I am looking a proof using Peano's axiom as I am working on the Natural Numbers, not the Real Numbers.
I need help proving "For each x,y,z in N, if $x<y$ then $x+z<y+z$
This must be done using Peano's axioms and the definitions of addition, multiplication, and ordering.
I have tried induction on z, and the base case works out but I run into an issue when it comes to the inductive part seeing as I am no longer dealing with equality.
I think induction is the correct way to approach this proof, however I become stuck once I reach a certain point in the proof.