So I inducted on n with a fixed m and I was told that it was good enough. But Im confused about one thing. How does this show that this works for all m aswell? Wont I have to also induct on m with a fixed n with a very similiar proof to this? I showed my prof this proof and they said the same thing: "I did take a look at your predicate and it seems to be for a fixed m. How do you show that the claim is true for all m? If you fix m in your predicate, you still need to be able to explain why the claim is true for all m >= 1"
Well, you prove that for any $m$, the following holds: $$ ll n, P'm(n)$$