If I have a proposition $P(n), n \in \{ 1,2,\ldots, m\}$ and I want to prove it by induction, can I proceed like this:
- Show that $P(1)$ is true
- Suppose that $P(n)$ is true for all $n \in \{1,2,\ldots,t\}$, where $t \in \mathbb{N}^*, t \le m-1$ and show that $P(t+1)$ is true.
This is something like complete induction on $\mathbb{N}$. It works? Thanks!