Lemma 2.2.3. For any natural numbers $n$ and $m$, $n + ( m{+\!+})$ = $(n+m){+\!+}$: Analysis 1, Terence Tao

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I'm self-learning real analysis using Terence Tao Analysis books. The book is very lucid. But in some cases Terence quickly go through the proof. And it becomes difficult for lesser mortals like us to understand. Lemma 2.2.3 is one of them. So, I'm requesting you guyz if you can help me understand this proof. The proof is the following:

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The portion, I'm not getting is:

But by definition of addition, $0 + (m{+\!+}) = m{+\!+}$ and $0 + m = m$, so both sides are equal to $m{+\!+}$ and are thus equal to each other.

Why $m{+\!+}$ & $m$ are equal. Similay, he is using similar logic in following section:

Similarly, we have $(n{+\!+} )+m = (n+m){+\!+}$ by the definition of addition, and so the right-hand side is also equal to $((n+m){+\!+}){+\!+}.$

These might be very simple logic. But I'm having hard time getting grasp of it. If someone can explain it, I'll be extremely grateful.

Thanks!

Best Answer

Tao never asserts that $m$ and $m{+\!+}$ are equal. What he asserts is that $0+(m{+\!+})=m{+\!+}$ and that $(0+m){+\!+}=m{+\!+}$. So, in particular, since both $0+(m{+\!+})$ and $(0+m){+\!+}$ are equal to $m{+\!+}$, $0+(m{+\!+})=(0+m){+\!+}$.

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