some problem in strong induction

hi

here is a problem i am trying to do.

where is nth Fibonacci number. following is my attempt to prove.

let

let m be arbitrary in N. so the goal is

where k runs in N.so let n be arbitrary in N. suppose

and

since we have to prove P(n), suppose

I have proven some base cases for n=0,1,2 so the next case is

now

so using inductive hypothesis, it follows that

we can also come up the following conclusion for n-1,

but I don't know what to do with this. I thought of another approach. since I assumed that

I can use this for the inductive hypothesis , but I can't be sure if

because since m is arbitrary in N,m could be zero, in which case the above inequality fails.

So what approach should I take here ?

Re: some problem in strong induction

I learned through some other forums that we have to have the condition

after that the proof is easy