Hey kkar.
As an alternative, have you tried expanding out the sequence using definitions for A_n+k in terms of all the ones less than n+k?
Define a sequence of numbers A_{i} by: A_{0} = 2, A_{n+1} = A_{n}/2 + 1/A_{n }(for n greater than or equal to 1). Prove that A_{n }less than or equal to sqrt(2) + 1/2^{n} for all n greater than or equal to 0. I think it's a safe bet that induction should be used here. I'm having trouble finding the inductive step though maybe just cause I'm not feeling particularly well today. Help is always appreciated.