Why did Terence Tao write Proposition 2.1.11 about mathematical induction in "Analysis I"?

Question

I am reading "Analysis I" by Terence Tao.

Axiom 2.5 (Principle of mathematical induction).
Let $P(n)$ be any property pertaining to a natural number $n$. Suppose that $P(0)$ is true, and suppose that whenever $P(n)$ is true, $P(n++)$ is also true. Then $P(n)$ is true for every natural number $n$.

Next he wrote this proposition:

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 $n$.

What is this proposition?
I cannot understand why Tao wrote this proposition.
Isn't it obvious from Axiom 2.5?

I cannot understand.

Answer 1

Okay, I looked up the text on line and

Tao will use the numbering system $x.y.z$ to highlight his main point even if they are just remarks or examples.

In this case, 2.1.11 is not actually a proposition. It is an example of instructions on how to do a proof by induction.

Axiom 2.5 says that principal of induction is what it is. "Proposition" 2.1.11 is instructions on how to use it.

So if I wanted to prove that all natural numbers taste like ice cream, I could do it by doing the following:

• First prove $0$ tastes like ice cream.
• Then prove that whenever $n$ tastes like ice cream that $n++$ tastes like ice cream.

Then I would have proven that all natural numbers taste like ice cream thanks to Axiom 2.5.

And I know how to prove it thanks to the instruction in "Proposition" 2.1.11.

.....

It's worth noting that the paragraph before "Proposition" 2.1.11 is:

The principle of induction gives us a way to prove that a prop-ertyP(n) is true for every natural numbern. Thus in the rest of this text we will see many proofs which have a form like this:

So this is not an actual proposition and proof but an outline of what a proposition and a proof by induction would look like.