Exercise.
The Fibonacci sequence ( ) is defined by 0= 1=1 and +2= + +1 .
(a) Prove that 2 for all . (Use induction on .)
(b) Say whether ( ) has a limit in and if it does, find this limit.
The sequence ( ) is defined by 0=1, 1=12 , and +2=2 +1+ 4 . Prove the following.
(a) 0 for all .
(b) =2- for all .
(c) ( ) is monotonic nonincreasing.
(d) Say whether ( ) has a limit in and if it does, find this limit.