I hate to say this, but...I'm stuck on an induction proof. I'm trying to prove that , where is the th Fibonacci number. I have it reduced to proving that for all . I have the rest of the proof.

So I tried proving it with induction.

It's trivial for , so I assumed . Then I tried to prove that .

Using an identity I know and have proved,

for some , but now I'm stuck.

Any hints?