MathNet 2009-part-04 — 175 problems (EN–VI) Question 106

The sequence $(u_n)$ is defined by conditions: $u_0 = 0$, $u_1 = \frac{1}{3}$ and $\frac{1}{2} u_n = \frac{1}{2}(u_{n+1} + u_{n-1})$. Prove that for all positive integers $n$ it holds $|u_n| \le 1$.