General Question about Induction

Question

If I prove a Proposition which I assume holds for all $n \in N $ and I show the statement is true for n=0 and then I want to show it for it n+1. Can I also use that the statement is true for n-1? Or do I have to add n=1 to the base case and then say that the statements hold for two consecutive numbers?

Answer 1

It sounds like you want to use $\varphi(n-1)\land\varphi(n)\to\varphi(n+1)$ as your inductive step, in which case yes, you need the first two values for $n$ in your base step. But depending on what you're trying to prove, you might find strong induction helps too.

Answer 2

Given that you have proved the truth of $P(0)$ and are trying to prove the truth of $P(n)$ then you can assume that all $P(i)$ are true for $0\le i\le n-1$.

The only "issue" arises with $n=1$ when you can, of course, only assume $P(0)$.