Question 112
The sequence $\{a_i\}_{i=1}^\infty$ is such that $a_1 \gt 0$ and $a_{n+1} = \frac{a_n}{1+a_n^2}$ for $n \ge 1$. Prove that:
a) $a_n \le \frac{1}{\sqrt{2n}}$ for $n \ge 2$;
b) there exists $n$ such that $a_n \gt \frac{7}{10\sqrt{n}}$.
a) $a_n \le \frac{1}{\sqrt{2n}}$ for $n \ge 2$;
b) there exists $n$ such that $a_n \gt \frac{7}{10\sqrt{n}}$.