Hi

here's problem i am trying to solve.

Suppose that is a real number and is an integer.

Prove that

I have to prove this using strong induction. Let

so I have to prove that

strong induction means I have to prove

so let n be arbitrary and suppose and since i have to prove

P(n) , lets suppose . Even though this is proof by strong induction, i will

prove some base cases.

If n=1 then by hypothesis

now let n=2 .

expanding the bracket , we have

so now consider n from 3 onwards. now

since n is now taken from 3 onwards,

letting k=n-1 in the inductive hypothesis, we can deduce that

now expanding, we get

now I will first prove that

we know that n-2 < n and since n is going from 3 onwards ,

from inductive hypothesis it follows that

and since is closed under addition , it follows that

since n is arbitrary

is the reasoning right ??