MathNet 2022-part-04 — 197 problems (EN–VI) Question 188

The sequence $(a_k)$ is given by $a_1 = \frac{1}{2}$ and $a_{n+1} = 1 - a_1 a_2 \cdots a_n$ for every $n \ge 1$. Prove that $a_{100} \gt 0.99$.