In my recent work, I have used the following double induction:
Let $ P(n,m) $ be a proposition about $n,m\in\mathbb{N}\cup\{0\}$. To show $P(n,m)$ is true for all $n\in\mathbb{N}\cup\{0\}$ and all $m\in\mathbb{N}\cup\{0\}$, I did:
Prove $P(0,m)$ is true for all $m\in\mathbb{N}\cup\{0\}$. (I can show directly for all $m\in\mathbb{N}\cup\{0\}$ is true, without induction on $m$).
Assume $P(n_0,m)$ is true for some $n_0\in\mathbb{N}\cup\{0\}$ and all $m\in\mathbb{N}\cup\{0\}$, show that $P(n_0+1,m)$ is true for all $m\in\mathbb{N}\cup\{0\}$.
Then, conclude that $P(n,m)$ is true for all $n,m\in\mathbb{N}\cup\{0\}$.
In the beginning, I also cast doubt on the logic of my own proof. Thanks to the answer from Mario Carneiro at:
that convince me a lot that my proof is right.
My Question:
Actually, I am not quite sure if this is a 'double induction' since I prove directly for any $m$ at inductive step instead of doing induction on it. Since I am a stranger to set theory so I really need an authoritative reference (i.e. books or papers) which clearly show this method with some simple examples of that. So that I can put my proof on my work. And, BTW, is there any term or special name for this kind of induction?
I really appreciate any help on that in advance.
