Given a sequence of rational numbers, defined inductively as follows:
s1 = 1, sn+1 = sn/2 + 1/sn, n>=1
prove that 1<=sn<=2 forall n>=1
The Attempt at a Solution
I've got the solution to this but I don't understand a certain part, I was hoping someone could explain it to me?
So Let P(n) be the statement 1<=sn<=2
P(1) is satisfied.
Suppose P(n) holds.
Then sn+1 = sn/2 + 1/sn >= 1/2 + 1/2 = 1, because sn/2>=1/2 and 1/sn<=1/2
The bold part is what i don't understand. Wouldn't that imply that sn >= 2?
Am I mistaken or is there maybe an error in this solution?
Thanks in advance.