# Doubt in Definition of Addition in Real- Analysis

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.

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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. – Epsilon Nov 19 '12 at 6:21

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.