Can we do the inductive step without using inductive hypothesis

Question

I wonder if in case I correctly performed first step of induction and next I proved it for n+1, this is still Mathematical Induction?

I know, that if I proved it without inductive hypothesis, I could not use induction at all, but I'm curious if it is proper solution if exercise told me to use it.

Answer 1

Yes, you can. The induction step amounts to a proof of : $P(n) \to P(n+1)$.

If you have a proof of $P(n+1)$ you can use the tautology : $\mathcal A \to (\mathcal B \to \mathcal A)$ to derive :

$P(n+1) \vdash P(n) \to P(n+1)$.