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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.

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Good question ; you were clear and provided your thoughts so that we can answer the precise question you are having trouble with. +1 –  Patrick Da Silva Nov 20 '12 at 6:16
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up vote 3 down vote accepted

You are correct that you are not addressing the entire "if and only if"

In order to show "if and only if" you must show that each of the two statements imply each other. In this case we need to show:

(1) $A \geq B$ implies $A + C \geq B + C$

&

(2) $A + C \geq B + C$ implies $A \geq B$

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+1 for leaving the proof to OP. That was precisely his question and you answered it. –  Patrick Da Silva Nov 20 '12 at 6:15
    
Ok. I'll finish the proof and post it. Thanks a lot ! –  Raghavendra Kumar Nov 20 '12 at 6:38
    
@devenware So this is my solution. Please tell me if it's right. I'm only proving the first part as I've already proved the second part. $A\geqB \Rightarrow A = B + m $ , where m belongs to whole numbers. From the above equation, we get, $\ A + C = B + C + m $ $\ \Rightarrow A + C \geq B + C $ –  Raghavendra Kumar Nov 20 '12 at 12:45
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@RaghavendraKumar Yes that seems like a good solution to me! –  Deven Ware Nov 20 '12 at 16:58
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