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I'm confused about why this proposition is true if this property is not true at all.

$Proposition$ 2.1.11.

A certain property $P(n)$ is true for every natural number n.

Proof. We use induction. We first verify the base case $n = 0$, i.e., we prove $P (0)$. (Insert proof of $P (0)$ here). Now suppose inductively that n is a natural number, and $P(n)$ has already been proven. We now prove $P$(n++). (Insert proof of $P(n++)$, assuming that $P(n)$ is true, here). This closes the induction, and thus $P(n)$ is true for all numbers n.

Thanks.

Mauro ALLEGRANZA
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廖倪徵
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  • Relevant - https://math.stackexchange.com/questions/3306914/why-did-terence-tao-write-proposition-2-1-11-about-mathematical-induction-in-an –  Sep 17 '19 at 13:09
  • It is only the "general sketch" of a proof by induction. See page 20 : "The principle of induction gives us a way to prove that a property P(n) is true for every natural number n. Thus in the rest of this text we will see many proofs which have a form like this: ... Of course we will not necessarily use the exact template, wording, or order in the above type of proof, but the proofs using induction will generally be something like the above form." – Mauro ALLEGRANZA Sep 17 '19 at 13:15

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