I am currently studying Analysis 1 by Terence Tao and in the proof of proposition 2.2.8 I saw author didn't use inductive hypothesis to complete the induction.
Proposition 2.2.8: If $a$ is a positive natural number, and $b$ is a natural number, then $a+b$ is positive.
Proof: We use induction on $b$. If $b=0$, then $a+b=a+0=a$, which is positive, so this proves the base case. Now suppose inductively that $a+b$ is positive. Then $a+(b++)=(a+b)++$, which cannot be zero by axiom 2.3, and is hence positive. This closes the induction. ∎
Notice:
Axiom 2.3 states that zero is not the successor of any natural number.
In the above proof $n++$ represents successor of $n$. In standard notations $S(n)$.
My Question: In the above proof Tao used axiom 2.3 to show that $(a+b)++$ can't be zero. But why he didn't used inductive hypothesis anywhere? I think it should be like this "Since $a+b$ is positive, by inductive hypothesis thus $(a+b)++$ also be positive that is successor of any positive natural number will be positive". Please correct me if i am wrong. Thank you.