# solution verification – How do I find the error within this Fibonacci Sequence proof that is trying to prove that f(5) = 4?

I am working on a problem in my textbook where I am given this proof dealing with Fibonacci numbers. The function $$f$$ is defined by $$f(0) = f(1) = 1$$ and for all $$ngeq 2$$, and $$f(n) = f(n-1) + f(n-2)$$. The following proof is trying to prove $$f(4) = 5$$:

begin{align*} f(4) &= 5\ f(3)+f(2) &= 5\ (f(2)+f(1))+f(2) &= 5\ 2f(2) + 1 &= 5\ 2f(2) &= 4\ 2(f(1) + f(0)) &= 4\ 2(1+1) &= 4\ 4 &= 4 end{align*}

I know that this proof is incorrect, but I’m having a hard time finding how it is incorrect and coming up with sufficient reasoning. Every time I look at it, I can’t seem to find a noticeable error. Can anyone give me some pointers and/or suggestions as to how this proof is incorrect? Any help is appreciated.