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.