Question 80
The sequence of Fibonacci numbers $F_0, F_1, F_2, \ldots$ is defined by $F_0 = F_1 = 1$ and $F_{n+2} = F_n + F_{n+1}$ for all $n \ge 0$. For example, we have
$$
F_2 = F_0 + F_1 = 2, \quad F_3 = F_1 + F_2 = 3, \quad F_4 = F_2 + F_3 = 5, \quad F_5 = 8.
$$
The sequence $a_0, a_1, a_2, \ldots$ is defined by
$$
a_n = \frac{1}{F_n F_{n+2}} \quad \text{for all } n \ge 0.
$$
Prove that for all $m \ge 0$ we have:
$$
a_0 + a_1 + a_2 + \cdots + a_m \lt 1.
$$
$$
F_2 = F_0 + F_1 = 2, \quad F_3 = F_1 + F_2 = 3, \quad F_4 = F_2 + F_3 = 5, \quad F_5 = 8.
$$
The sequence $a_0, a_1, a_2, \ldots$ is defined by
$$
a_n = \frac{1}{F_n F_{n+2}} \quad \text{for all } n \ge 0.
$$
Prove that for all $m \ge 0$ we have:
$$
a_0 + a_1 + a_2 + \cdots + a_m \lt 1.
$$