I'm confused with the logic of a problem. I'm trying to prove a statement of the form $P(n,k)$ where $1 \leq k\leq n$. Based on the problem it seems natural to use (strong) induction on $n$. Proving the base case is easy, and then when you want to prove $$P(n,k) \implies P(n+1,k) $$ you need to use $P(n-1,k)$ and $P(n-1,k-1)$, so I though that I needed to do something like double induction on $n$ and $k$. But these wouldn't really work because $k$ is bounded by $n$. Is there a way that I get get around this?
I hope to get some help only with just this information. I really don't want to post the problem because it is a homework problem and I don't want unnecessary help.