Solution
= Solution
The <Fibonacci number>[Fibonacci recurrence] gives
$$
a_{n+1}=\frac{F_{n+2}}{F_{n+1}}
=1+\frac{F_n}{F_{n+1}}=1+\frac1{a_n}.
$$
The map $x\mapsto1+1/x$ is strictly decreasing for $x>0$. Since $a_3=2\geq a_1=1$, applying this decreasing map reverses each inequality and proves by <mathematical induction> that
$$
(-1)^na_{n+2}\leq(-1)^na_n.
$$
Taking even $n$ shows $a_{2n+2}\leq a_{2n}$, so $(a_{2n})$ is a <monotone sequence>[decreasing sequence].