IR-TST 2023 Problem 8

Suppose $\frac{1}{2} < s < 1$ . An insect flying on $[0,1]$ . If it is on point $a$ , it jump into point $a\times s$ or $(a-1) \times s +1$ . For every real number $0 \le c \le 1$ , Prove that insect can jump that after some jumps , it has a distance less than $\frac {1}{1402}$ from point $c$ .

Proposed by Navid Safaei