Question 5
Let $a_0, a_1, \dots, a_N$ be real numbers satisfying $a_0 = a_N = 0$ and
$$
a_{i+1} - 2a_i + a_{i-1} = a_i^2
$$
for $i = 1, 2, \dots, N-1$. Prove that $a_i \le 0$ for $i = 1, 2, \dots, N-1$.
$$
a_{i+1} - 2a_i + a_{i-1} = a_i^2
$$
for $i = 1, 2, \dots, N-1$. Prove that $a_i \le 0$ for $i = 1, 2, \dots, N-1$.