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 $n\geq 2$, 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.