Question 100
Let $(F_k)_{k\geqslant0}$ be the sequence defined by $F_0=0,F_1=1$ and $F_{k+2}=F_k+F_{k+1}$ for every integer $k\geqslant0$. Let $n\geqslant1$ be an integer. Prove that there are exactly $F_{n+1}$ ways to order $1,2,\ldots,n$ into an $n$-tuple $(a_1,a_2,\ldots,a_n)$ satisfying
$$a_1\leqslant2a_2\leqslant3a_3\leqslant\cdots\leqslant na_n.$$
$$a_1\leqslant2a_2\leqslant3a_3\leqslant\cdots\leqslant na_n.$$