The problem:
Let $F_n$ be the nth term of the Fibonacci sequence:
$F_0 = 0$
$F_1 = 1$
$F_n = F_{n-1} + F_{n-2}$ for $n\geq2$
Prove that $\sum_{i=1}^{n} F_i^2 = F_nF_{n+1}$ for all $n \in\mathbb{N}$
My attempt to prove this using the induction hypothesis is:
1) With $n = 1$, the equation holds true: $F_{1}^2 = F_{1}F_{n+1}$ because $1^2 = 1*1$.
2) Now we have to prove that $\sum_{i=1}^{n} F_i^2 = F_nF_{n+1} \implies \sum_{i=1}^{n+1} F_i^2 = F_{n+1}F_{n+2}$
We know that $\sum_{i=1}^{n+1} F_i^2 = \sum_{i=1}^{n} F_i^2 + F_{n+1}^2$, and if we assume that the antecedent is true we get:
$\sum_{i=1}^{n+1} F_i^2 = F_{n}F_{n+1} + F_{n+1}^2$
If we replace this last equation in the consequent we get:
$F_{n}F_{n+1} + F_{n+1}^2 = F_{n+1}F_{n+2}$
Finally, if we divide both sides by $F_{n+1}$ we end up with the Fibonacci recurrence equation:
$F_{n+2} = F_{n+1} + F_{n}$
We know this holds for all $n\in\mathbb{N}$, because $F_n$ is defined as the nth term of the Fibonacci sequence by the premises of the problem. Therefore we have proven (2).
Thus, by the Principle of Mathematical Induction: $\sum_{i=1}^{n} F_i^2 = F_nF_{n+1}$ for all $n \in\mathbb{N}$
QUESTION: Is this proof correct? And if not, where is the mistake?