 Solution

The Fibonacci recurrence givesThe map $x\mapsto 1+1/x$ is strictly decreasing for $x>0$. Since $a_3=2\ge a_1=1$, applying this decreasing map reverses each inequality and proves by mathematical induction thatTaking even $n$ shows $a_{2n+2}\le a_{2n}$, so $(a_{2n})$ is a decreasing sequence.