PL-OM 2024 Problem 5

We are given an integer $n \ge 2024$ and a sequence $a_1,a_2,\dots,a_{n^2}$ of real numbers satisfying $$\vert a_k-a_{k-1}\vert \le \frac{1}{k} \text{and} \vert a_1+a_2+\dots+a_k\vert \le 1$$ for $k=2,3,\dots,n^2$ . Show that $\vert a_{n(n-1)}\vert \le \frac{2}{n}$ . Note: Proving $\vert a_{n(n-1)}\vert \le \frac{75}{n}$ will be rewarded partial points.