Question 122
Given integer $n \gt 2$, suppose positive real numbers $a_1, a_2, \dots, a_n$ satisfy $a_k \le 1$, $k = 1, 2, \dots, n$.
Let $A_k = \frac{a_1 + a_2 + \dots + a_k}{k}$, $k = 1, 2, \dots, n$.
Prove $\left| \sum_{k=1}^n a_k - \sum_{k=1}^n A_k \right| \lt \frac{n-1}{2}$.
Let $A_k = \frac{a_1 + a_2 + \dots + a_k}{k}$, $k = 1, 2, \dots, n$.
Prove $\left| \sum_{k=1}^n a_k - \sum_{k=1}^n A_k \right| \lt \frac{n-1}{2}$.