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From Tao, Analysis I, pp. 77-78:

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I don't quite understand why the second paragraph is necessary.

In the first paragraph, we use an earlier result---trichotomy of natural numbers: if $a$ and $b$ are natural numbers, then exactly one of (i) $a>b$; (ii) $a=b$; or (iii) $a<b$ is true. We now show that (i) $\implies$ (b), (ii) $\implies$ (a), and (iii) $\implies$ (c). Isn't this sufficient to complete the proof?

Why is the second paragraph necessary?

Am I missing something?

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    In the first paragraph it is shown that for every integer $x$ it is always true one of the statements (a), (b) or (c), but it could be the case that altough as you say to (i) corresponds (b), by another way of thinking about it, you could get that to (i) also corresponds (c). That's what in the second paragraph is proven to be impossible. – Eparoh Dec 26 '21 at 10:43
  • @Eparoh: We've already shown that (i) $\implies$ (b), so how could (i) correspond to (c)? –  Dec 27 '21 at 02:32
  • Btw why is the - operator that long? – Hermis14 Dec 27 '21 at 03:12
  • @Hermis14: Read the previous few pages (available through Google Books link given above) –  Dec 27 '21 at 03:26

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Yes, the previous trichotomy statement is enough to complete the proof, assuming that you have already proved that, if an integer $a$ is different from $0$, then $a\in\Bbb N$ or $-a\in\Bbb N$. But Tao decided to use another approach, which is neither better nor worst than that one.

José Carlos Santos
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