Question 44
Let $\frac{1}{2} \lt s \lt 1$. An insect jumps on points in the interval $[0, 1]$. If the insect is on point $a$, it can jump to either $s \times a$ or $(a - 1) \times s + 1$. Prove that for any arbitrary point $c$ in the interval $[0, 1]$, the insect can jump in such a way that after a few steps, it will be at a distance less than $\frac{1}{1402}$ from $c$.