= Solution
The stated formula is <Fibonacci determinant identity>[Cassini's identity]. It holds for $n=1$, and the recurrence gives
$$
F_{n+3}F_{n+1}-F_{n+2}^2
=F_{n+1}^2-F_{n+2}F_n,
$$
which is the negative of the expression at index $n$. This proves the identity by induction.
Consequently
$$
a_{n+1}-a_n
=\frac{F_{n+2}F_n-F_{n+1}^2}{F_{n+1}F_n}
=\frac{(-1)^{n+1}}{F_{n+1}F_n}.
$$
The positive <Fibonacci number>[Fibonacci numbers] tend to <infinity>, so this difference tends to zero.
Back to article page
