I'm starting R.P Burn "Number and Functions, Steps into Analysis". In Chapter 2, the basic properties of inequalities are derived, and question 8 asks to prove using the fact that positive numbers are closed under addition that $a<b \Rightarrow a+c<b+c$.
The solution presented in the book is: $a<b \Rightarrow 0<b-a \Rightarrow 0 < (b+c) - (a+c) \Rightarrow a+c < b+c$
Where is the property that the positive numbers are closed under addition used in this proof? It seems to me like it's just adding $+c$ and $-c$ to both sides of the inequality, which works but I don't see how it relies on the closure property.