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I am going through Analysis V.1, Terence Tao. In his definition of Addition, screenshot given below, how did he deduce that (N++)+M := (N+M)++ ?? I am not able to understand the steps.

Definition of Addition Definition 2.2.1, Volume 1 Analysis, Terence Tao

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Thats a recursive definition of addition of natural numbers. If you know what n+m is, then you define inductively (on n) what $(n++)+m$ is. You could also view it as function $f_m:\mathbb N \rightarrow \mathbb N:m \mapsto n+m$. Then you define $f(0)=m$ and given that value you definie $f(n++)$. Thus $f(n++):=f(n)++$, which is a natural way do to so. –  André Nov 19 '12 at 6:21

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You have to remember that in developing formal rules for addition, we are taking the properties in which we are already naturally familiar and changing them into a rigorous definition.

Informally, we perceive $n++$ as $n+1$. Therefore the recursive definition simply says that $$(n+1) + m = (n+m)+1$$ You must remember that we already have in mind what properties we want addition to have and that we are simply providing a definition.

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I have understood the notation. I have not understood how he proved the equation shown above. More precisely, how did he 'use' the concept of recursion and the knowledge upto that point in the book to prove the above identity/Lemma. –  Raghavendra Kumar Nov 19 '12 at 12:59
    
He doesn't prove it. He defines addition as the above recursive rule. Up to this point, there is not such thing as addition, so there is nothing to prove. –  EuYu Nov 19 '12 at 13:02

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