I am an engineer and just started getting into pure mathematics (to get a complete understanding of the workings of calculus. I cannot 'use' calculus unless I know how it works), so please bear with me.
In the book, Analysis - Volume I, by Terence Tao, he gives the following proposition
Prove $A\geq B$ if and only if $A+C \geq B+C$ where $A,B,C$ belong to whole numbers.
My solution is : $A+C \geq B+C$ implies $A+C = B+C + m$ ( where $m$ is a whole number )
We can cancel $C$ to get $A = B + m$, which by the definition implies that $A \geq B$.
My question is I am not sure if my proof is complete. I am not clear if I am addressing the "if and only if" part of the proposition.