MathNet 2026 — Part 05 Question 144

Suppose $0 \le x \lt 1$ and let $a_n = \frac{1-x^n}{n(1-x)}$ for all integers $n \ge 1$. Prove that
$$
|a_{n+1} - a_n| \le \frac{1}{n+1}, \quad n = 1, 2, 3, \dots
$$