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