NL-NWO 2019 Problem 4

The sequence of Fibonacci numbers $F_0, F_1, F_2, . . .$ is defined by $F_0 = F_1 = 1$ and $F_{n+2} = F_n+F_{n+1}$ for all $n > 0$ . For example, we have $F_2 = F_0 + F_1 = 2, F_3 = F_1 + F_2 = 3, F_4 = F_2 + F_3 = 5$ , and $F_5 = F_3 + F_4 = 8$ . The sequence $a_0, a_1, a_2, ...$ is defined by $a_n =\frac{1}{F_nF_{n+2}}$ for all $n \ge 0$ . Prove that for all $m \ge 0$ we have: $a_0 + a_1 + a_2 + ... + a_m < 1$ .