Yes, that's right. I would only say, "Since is closed under subtraction..." since you deduce from and .
here's problem i am trying to solve.
Suppose that is a real number and is an integer.
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 ??