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?
• You can either assume that the statement is true for $n$ and prove it is true for $n+1$ of you can assume it is true for all natural number $\leq n$ and prove it true for $n+1$. The two methods are equivalent. – saulspatz Oct 9 '19 at 21:09
• "Can I also use that the statement is true for n-1?" -> I'm not sure what you mean .. are you trying to prove that $P(n+1)$ on the basis of $P(n-1)$ (in addition to $P(n)$)? Or are you trying to go from $P(n-1)$ to $P(n)$ instead of going from $P(n)$ to $P(n+1)$? – Bram28 Oct 9 '19 at 21:13
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.
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 $$P(0)$$.