The Fibonacci numbers $F_{n}$ are defined by $F_{1}=1, F_{2}=1, F_{n+2}=F_{n+1}+F_{n}(n \geqslant 1)$. Let $a_{n}=F_{n+1} / F_{n}$ be the ratio of successive Fibonacci numbers.

(i) Show that $a_{n+1}=1+1 / a_{n}$. Hence prove by induction that

$$(-1)^{n} a_{n+2} \leqslant(-1)^{n} a_{n}$$

for all $n \geqslant 1$. Deduce that the sequence $a_{2 n}$ is monotonically decreasing.

(ii) Prove that

$$F_{n+2} F_{n}-F_{n+1}^{2}=(-1)^{n+1}$$

for all $n \geqslant 1$. Hence show that $a_{n+1}-a_{n} \rightarrow 0$ as $n \rightarrow \infty$.

(iii) Explain without detailed justification why the sequence $a_{n}$ has a limit.